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ELEMENTARY LESSONS IN LOGIC 



ELEMENTARY LESSONS 
IN LOGIC: 

DEDUCTIVE AND INDUCTIVK 




WITH COPIOUS QUESTIONS AND EXAMPLES^ 

AND 

A VOCABULARY OF LOGICAL TERMS. 

Wj^'' STANLEY JEVONS, M.A. 



PROFESSOR OF LOGIC IN OWENS COLLEGE, MANCHESTEa. 

NE W EDITION. 

MACMILLAN AND CO. 
1882. 



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PREFACE. 

In preparing these Lessons I have attempted to 
show that Logic, even in its traditional form, can be 
made a highly useful subject of study, and a powerful 
means of mental exercise. With this view I have 
avoided the use of superfluous technical terms, and 
have abstained from entering into questions of a 
purely speculative or metaphysical character. For 
the puerile illustrations too often found in works on 
Logic I have generally substituted examples drawn 
from the distinct objects and ideas treated in the 
natural and experimental sciences; and in this and 
other respects have aimed at rendering these Lessons 
a suitable companion to a series of science school- 
books. 



vi PREFACE, 

Logic is not only an exact science, but is the 
most simple and elementary of all sciences ; it ought 
therefore undoubtedly to find some place in every 
course of education. The relations of propositions 
and the forms of argument present as precise a sub- 
ject of instruction and as vigorous an exercise of 
thought, as the properties of geometrical figures, or 
the rules of Algebra. Yet every school-boy is made 
to learn mathematical problems which he will never 
employ in after life, and is left in total ignorance of 
those simple principles and forms of reasoning which 
will enter into the thoughts of every hour. Logic 
should no longer be considered an elegant and learn- 
ed accomplishment; it should take its place as an 
indispensable study for every well-informed person. 
These Lessons I trust will introduce to the science 
many who have not leisure or inclination to read more 
elaborate treatises, and many who would not be at- 
tracted by the numerous but somewhat dry and brief 
compendiums published in past years. 

It is desirable that Lessons in Logic should be 
made the basis of many exercises, and for this pur- 
pose I have supplied abundance of questions and 
examples at the end of the book, some of which are 
selected from the examination papers of the Oxford, 



PREFACE, \\\ 

London, and Edinburgh Universities. In my own 
classes I have constantly found that the working and 
solution of logical questions, the examination of argu- 
ments and the detection of fallacies, is a not less 
practicable and useful exercise of mind than is the 
performance of calculations, and the solution of pro- 
blems in a mathematical class. 

Except in a few places, where special notice is 
given, I have abstained from putting forward any 
views not commonly accepted by teachers of logic; 
and I have throughout devoted more attention to 
describing clearly and simply the doctrines in which 
logicians generally agree, than discussing the points 
in which there is a difference of opinion. The recent 
logical discoveries of Sir W. Hamilton, Archbishop 
Thomson, Prof, de Morgan, and especially the late 
Prof. Boole, cannot yet be fully adopted in an ele- 
mentary work, but I have attempted to give a clear 
notion of the results to which they inevitably lead. 

In the latter Lessons which treat of Induction I 
have generally followed Sir John Herschel, Dr Whewell 
and Mr J. S. Mill, as the recognised authorities on the 
subject. These Lessons in fact may be regarded as 
an easy introduction to some of the most import?'*.: 
parts of Mr Mill's treatise on Logic. 



viii PREFACE. 

At the end of almost every Lesson will be found 
references to the works in which the student will most 
profitably continue his reading of the subject treated, 
so that this little volume may serve as a guide to a 
more extended course of study. 



TABLE OF CONTENTS. 



LBSSON rAGS 

I. Definition and Sphere of the Science *,. i 

IL The Three Parts of Logical Doctrine 9 

TERMS. 

III. Terms, and their various Kinds i6 

IV« Of the Ambiguity of Terms r. "27 

V. Of the twofold meaning of terms — in Extension 

and Intension 37 

VI. The Growth of Language •... 44 

VIL Leibnitz on Knowledge 53 

PROPOSITIONS. 

VIII, Kinds of Propositions ...4 « 60 

IX. The Opposition of Propositions ^ 71 

X. Conversion of Propositions, and Immediate In- 
ference 81 

XI. Logical Analysis of Sentences , 88 

XIL The Predicables, Division, and Definition 98 

XIII. Pascal and Descartes on Method 11 1 



K. 



TABLE OF CONTENTS. 



SYLLOGISM. 

LESSON ^ FAGB 

XIV. The Laws of Thought 117 

XV. The Rules of the Syllogism 126 

XVI. The Moods and Figures of the Syllogism , 135 

XVIL Reduction of the Imperfect Figures 144 

XVIII. Irregular and Compound Syllogisms ...^. 152 

XIX. Of Conditional Arguments ..., 160 

FALLACIES. 

XX. Logical Fallacies » *» 169 

XXI. Material Fallacies ^ 176 

RECENT LOGICAL VIEWS. 

XXIL The Quantification of the Predicate 183 

XXIII. Boole's System of Logic 191 

METHOD. 

XXIV. Of Method, Analysis^ and S3mthesis ^.^ 201 



INDUCTION. 

XXV. Perfect Induction and the Inductive Syllogism 110 

XXVI. Geometrical and Mathematical Induction, Ana- 

logy, and Example .^ 2i8 

XXVII. Observation and Experiment *... 228 

XXVIII. Methods of Induction *-. 239 

XXIX, Methods of Quantitative Induction 247 



TABLE OF CONTENTS. xi 

LESSON PAOB 

XXX. Empirical and Deductive Methods 255 

XXXI. Explanation, Tendency, Hypothesis, Theory 

and Fact 264 

SUBSIDIARIES OF INDUCTION. 

XXXII. Classification, and Abstraction 276 

XXXIII. Requisites of a Philosophical Language 287 



Questions and Exercises *....'. 296 

Examples of Terms 297 — 299 

Examples of Propositions 303 

Examples of Arguments *....... 312, 315 

Index .....,.......«....*«:>.,.,• 332 



INTRODUCTION. 
LESSON I. 

DEFINITION AND SPHERE OF THE SCIENCE. 

Logic may be most briefly defined as the Science of 
Reasoning. It is more commonly defined, however, as the 
Science of the Laws of Thought, and some logicians think 
it desirable to specify still more accurately that it is the 
Science of the Formal, or of the Necessary Laws of 
Thought. Before these definitions can be of any real 
use to us we must come to a clear understanding as to 
the meaning of the expressions ; and it will probably 
appear that there is no great difference between them. 

By a Law of Thought we mean a certain uniformity or 
agreement which exists and must exist in the modes in 
which all persons think and reason, so long as they do not 
make what we call mistakes, or fall into self-contradiction 
and fallacy. The laws of thought are natural laws with 
which we have no power to interfere, and which are of 
course not to be in any way confused with the artificial laws 
of a country, which are invented by men and can be altered 
by them. Every science is occupied in detecting and 
describing the natural laws which are inflexibly observed 

I 



2 DEFINITION AND SPHERE [LESS. 

by the objects treated in the Science. The science of 
astronomy investigates the uniform or similar way in 
which the heavenly bodies, and in fact all material sub- 
stances, tend to fall towards each other as a stone falls 
towards the earth, or to move round each other under 
the influence of this tendency. The universal law of 
gravitation is thus the natural law or uniformity treated 
in physical astronomy. 

In chemistry the law of equivalent proportions de- 
scribes the well ascertained fact that each chemical 
substance enters into combination with every other che- 
mical substance only in certain definite proportions ; as 
when exactly eight parts by weight of oxygen unite with 
one part of hydrogen to form water, or sixteen parts of 
oxygen and six parts of carbon unite to form carbonic 
acid in the ordinary burning of a flame or fire. When* 
ever we can detect uniformities or similarities we so far 
create science and arrive at natural laws. But there may 
be, and are, many things so fickle, complicated, and 
uncertain, that we can never be sure we have detected 
laws that they will imiformly obey ; in such cases no 
science, in the proper sense of the word, is possible. 
'I'here is no such thing, for instance, as a real science of 
human character, because the human mind is too variable 
and complicated a subject of investigation. There are 
no two persons so much alike that you may be sure of 
one acting in all circumstances as the other would; it 
thus becomes impossible to arrange persons in classes so 
that all who are in the same class shall act uniformly in 
the same manner in any given circumstances. 
•sJ But there is a science of human reason or thought 
/apart from the many other acts of mind which belong to 
human character, because there are modes in which all 
persons do uniformly think and reason, and must think 
and reason. Thus if two things are identical with a third 



1.1 OF THE SCIENCE, 3 

common thing they are identical with each other. This 
is a law of thought of a very simple and obvious charac- 
tei, and we may observe concerning it, — 

1. That all people think in accordance with it, and 
agree that they do so as soon as they understand its 
meaning. 

2. That they think in accordance with it whatever 
may be the subject about which they are thinking. 

Thus if the things considered are — 

London, 

The Metropolis, 

The most populous city in Great Britain, 
since "the Metropolis is identical with London," and 
" London is identical with the most populous city in 
Great Britain," it follows necessarily in all minds that 
" the metropolis is identical with the most populous city 
in Great Britain." 

Again, if we compare the three following things — 

Iron, 

The most useful metal. 

The cheapest metal, — 

and it be allowed that " The most useful metal is Iron^" 
and " Iron is the cheapest metal," it follows necessarily 
in all minds that "the most useful metal is the cheapest" 
We here have two examples of the general truth that 
things identical with the same thing are identical with 
each other ; and this we may say is a general or necessary 
form of thought and reasoning. 

Compare, again, the following three things, — 

The earth, 

Planets, 

Bodies revolving in elliptic orbits. 

We cannot say, as before, that "the earth is identical 
with the planets;" it is identical only with one of the 

I — 2 



4 DEFINITION AND SPHERE [less. 

planets, and we therefore say that " it is a planet." Simi- 
larly we may say that " the planets are bodies revolving 
in elliptic orbits," but only a part of the whole number 
so revolving. Nevertheless it follows that if the earth is 
among the planets, and the planets among bodies re- 
volving in elliptic orbits, then the earth is among the 
latter. 

A very elementary knowledge of chemistry enables us 
to argue similarly concerning the following ; — 

Iron, 

Metals, 

Elementary substances. 

Iron is one of the metals, and metals are elements or 
simple undecomposable substances, in the sense of being 
among them or a part of them, but not as composing the 
whole. It follows necessarily that " Iron is one of the 
elementary substances." We have had then two exam- 
ples of a fixed and necessary form of thought which is 
necessary and true whatever the things may be to which 
it is applied. The form of argument may be expressed in 
several different ways, and we shall have to consider it 
minutely in the lessons on the syllogism ; we may express 
it, for instance, by saying that "part of a part is part 
of the whole." Iron is part of the class of metals, which 
is part of the class of elements: hence iron is part of 
the class of elements. 

vj If I now introduce another definition of Logic and 
say that it is "the science of the necessary forms of 
thought," the reader will I hope clearly apprehend the 
meaning of the expression "necessary forms of thought." 
A form is something which may remain uniform and 
unaltered, while the matter thrown into that form may be 
varied. Medals struck from the same dies have exactly 
the same form, but they may be of various matter, aa 



L] OF THE SCIENCE. ^-.5 

bronze, copper, gold or silver. A building of exactly the 
same form might be constructed either of stone or bricks ; 
furniture of exactly similar shape may be made of oak, 
mahogany, walnut wood, etc. Just as we thus familiarly 
recognize the difference of form and substance in common 
tangible things, so we may observe in Logic, that the 
form of an argument is one thing, quite distinct from the 
various subjects or matter which may be treated in that 
form. We may almost exhibit to the eye the form of 
reasoning to which belong our two latter arguments, as 
follows : — 

•• (Y) •. - 

If within the three pairs of brackets, marked respect- 
ively Xy Y and Z we place three names, such that the 
one in place of X may be said to come under that in Y, 
and that in Y under that in Z, then it necessarily follows 
that the first {X) comes under the last {Z), 
V Logic, then, is the science occupied in ascertaining 
and describing all the general forms of thought which we 
must employ so long as we reason validly. These forms 
are very numerous, although the principles on which they 
are constructed are few and simple. It will hence appear 
that logic is the most general of all the sciences. Its 
aid must be more often required than the aid of any other 
science, because all the particular sciences treat portions 
only of existing things, and create very different and 
often unconnected branches of knowledge. But logic 
treats of those principles and forms of thought which 
must be employed in every branch of knowledge. It 
treats of the very origin and foundations of knowledge 
itself; and though it is true that the logical method em- 
ployed in one science may differ somewhat from that em- 



6 DEFINITION AND SPHERE [less. 

ployed in another science, yet whatever the particular 
form may be, it must be logical, and must conform to the 
laws of thought. There is in short something in which 
aU sciences must be similar ; to which they must con- 
form so long as they maintain what is true and self- 
consistent ; and the work of logic is to explain this 
common basis of all science. 

One name which has been given to Logic, namely the 
Science of Sciences, very aptly describes the all extensive 
power of logical principles. The cultivators of special 
branches of knowledge appear to have been fully aware 
of the allegiance they owe to the highest of the sciences, 
for they have usually given names implying this allegi- 
ance. The very name of logic occurs as part of nearly 
all the names recently adopted for the sciences, which are 
often vulgarly called the " ologies," but are really the 
"logics," the "o" being only a connecting vowel or part 
of the previous word. Thus geology is logic applied to 
explain the formation of the earth's crust ; biology is logic 
applied to the phenomena of life ; psychology is logic 
applied to the nature of the mind ; and the same is the 
case with physiology, entomology, zoology, teratology, 
morphology, anthropology, theology, ecclesiology, thalat- 
toiogy, and the resf^. Each science is thus distinctly 
confessed to be a special logic. The name of logic itself 
IS derived from the common Greek word \oyos^ which 
usually means word, or the sign and outward manifesta- 
tion of any inward thought. But the same word was also 
used to denote the inward thought or reasoning of which 
words are the expression, and it is thus probably that later 
Greek writers on reasoning were led to call their science 



* Except Philology, which is differently formed, and means 
the love or study of words ; the name of this science, if forr»e»J 
upon the same plan, would be logology. 



L] OF THE SCIENCE. i 

iiriaTTijxr) XoyiKi], or logical science ; also rex^rj \oyLKrj, or 
logical art. The adjective XoyiKi], being used alone, soon 
came to be the name of the science, just as Mathematic, 
Rhetoric, and other names ending in **ic" were ori- 
ginally adjectives but have been converted into substan- 
tives. 

Much discussion of a somewhat trifling character has 
arisen upon the question whether Logic should be con- 
sidered a science only, an art only, or both at the same 
time. Sir W. Hamilton has even taken the trouble to 
classify almost all the writers on logic according as they 
held one opinion or the other. But it seems substan- 
tially correct and sufficient to say, that logic is a science 
in so far as it merely investigates the necessary princi- 
ples and forms of thought, and thus teaches us to under- 
stand in what correct thinking consists; but that it be- 
comes an art when it is occupied in framing rules to assist 
persons in detecting false reasoning. A science teaches us 
to know and an art to do, and all the more perfect sciences 
lead to the creation of corresponding useful arts. As- 
tronomy is the foundation of the art of navigation on the 
ocean, as well as of the arrangement of the calendar and 
chronology. Physiology is the basis of the art of medi- 
cine, and chemistry is the basis of many useful arts. 
Logic has similarly been considered as the basis of an art 
of correct reasoning or investigation which should teach 
the true method to be observed in all sciences. The cele- 
brated British logician Duns Scotus, who lived in the 13th 
century, and called logic the Science of Sciences, called it 
also the Art of Arts, expressing fully its preeminence. 
Others have thus defined it — " Logic is the art of direct- 
ing the reason aright in acquiring the knowledge of 
things, for the instruction both of ourselves and others." 
Dr Isaac Watts, adopting this view of logic, called his 
well-known work '^ the Art of Thinking." 



8 DEFINITION AND SPHERE [less. 

It may be fairly said however that Logic has more 
the form of a science than an art for this reason — all 
persons necessarily acquire the faculty and habit of rea- 
soning long before they even know the name of logic. 
This they do by the natural exertion of the powers of 
mind, or by constant but unconscious imitation of others. 
They thus observe correctly but unconsciously the prin- 
ciples of the science in all very simple cases ; but the con- 
tradictory opinions and absurd fallacies which are put 
forth by uneducated persons shew that this unaided ex- 
ercise of mind is not to be trusted when the subject of 
discussion presents any difficulty or complexity. The 
study of logic then cannot be useless. It not only 
explains the principles on which every one has often 
reasoned correctly before, but points out the dangers 
which exist of erroneous argument. The reasoner thus 
becomes consciously a correct reasoner and learns con- 
sciously to avoid the snares of fallacy. To say that 
men can reason well without logical science is about as 
true as to say that they can live healthily without medi- 
cine. So they can— as long as they are healthy ; and so 
can reasoners do without the science of reasoning — as long 
as they do reason correctly ; but how many are there that 
can do so ? As well might a man claim to be immortal 
in his body as infallible in his mind. 

And if it be requisite to say a few words in defence of 
Logic as an art, because circumstances in the past his- 
tory of the science have given rise to misapprehension, 
can it be necessary to say anything in its praise as a 
science 1 Whatever there is that is great in science or in 
art or in literature, it is the work of intellect. In bodily 
form man is kindred with the brutes, and in his perish- 
able part he is but matter. It is the possession of con- 
scious intellect, the power of reasoning by general notions 
that raises him above all else upon the earth; and who 



II.] OF THE SCIENCE. 9 

can say that the nature and procedure of this intellect is 
not almost the highest and most interesting subject of 
study in which we can engage? In vain would any 
one deny the truth of the favourite aphorism of Sir W. 
Hamilton — 

In the world there is nothing great but man. 
In man there is nothing great but mind. 



LESSON II. 
THE THREE PARTS OF LOGICAL DOCTRINE. 

It has been explained in the previous lesson that Logic 
is the Science of Reasoning, or the Science of those Ne- 
cessary Laws of Thought which must be observed if we 
are to argue consistently with ourselves and avoid self- 
contradiction. Argument or reasoning therefore is the 
stiictly proper subject before us. But the most conve- 
nient and usual mode of studying logic is to consider first 
the component parts of which any argument must be 
made up. Just as an architect must be acquainted with 
the materials of a building, or a mechanic with the ma- 
terials of a machine, before he can pretend to be ac- 
quainted with its construction, so the materials and in- 
struments with which we must operate in reasoning are 
suitably described before we proceed to the actual forma 
of argument. 

If we examine a simple argument such as that givea 
in the last lesson, thus- 
Iron is a metal, 
Every metal is an element, 
Therefore Iron is an element,— 



lo THE THREE PARTS OF [less. 

we see that it is made up of three statements or asser- 
tions, and that each of these contains, besides minor 
words, two nouns substantive or names of things, and the 
verb " is." In short, two names, or terms, when connected 
by a verb, make up an assertion or proposition; and 
three such propositions make up an argument, called in 
this case a syllogism. Hence it is natural and conve- 
nient first to describe terms, as the simplest parts ; next 
to proceed to the nature and varieties of propositions 
constructed out of them, and then we shall be in a posi- 
tion to treat of the syllogism as a whole. Such accord- 
ingly are the three parts of logical doctrine. 

But though we may say that the three parts of logic 
are concerned with terms, propositions, and syllogisms, 
it may be said with equal or greater truth that the acts of 
mind indicated by those forms of language are the real 
subject of our consideration. The opinions, or rather 
perhaps the expressions, of logicians have varied on this 
point. Archbishop Whately says distinctly that logic is 
entirely conversant about language ; Sir W. Hamilton, Mr 
Mansel, and most other logicians treat it as concerned 
with the acts or states of mind indicated by the words ; 
while Mr J. S. Mill goes back to the things themselves 
concerning which we argue. Is the subject of logic, then, 
language, thought, or objects? The simplest and truest 
answer is to say that it treats in a certain sense of all 
three. Inasmuch as no reasoning process can be ex- 
plained or communicated to another person without 
words, we are practically limited to such reasoning as is 
reduced to the form of language. Hence we shall always 
be concerned with words, but only so far as they are the 
instruments for recording and referring to our thoughts. 
The grammarian also treats of language, but he treats it 
as language merely, and his science terminates with the 
description and explanation of the forms, varieties, and 



11.1 LOGICAL DOCTRINE. ii 

relations of words. Logic also treats of language, but 
only as the necessary index to the action of mind. 

Again, so long as we think correctly we must think of 
things as they are ; the state of mind within us must 
correspond with the state of thmgs without us whenever 
an opportunity arises for comparing them. It is im- 
possible and inconceivable that iron should prove not to 
be an elementary substance, if it be a metal, and every 
metal be an element. We cannot suppose, and there is 
no reason to suppose, that by the constitution of the 
mind we are obliged to think of things differently from 
the manner in which they are. If then we may assume 
that things really agree or differ according as by correct 
logical thought we are induced to believe they will, it 
does not seem that the views of the logicians named are 
irreconcileable. We treat of things so far as they are the 
objects of thought, and we treat of language so far as it is 
the embodiment of thought. If the reader will bear this 
explanation in mind, he will be saved from some per- 
plexity when he proceeds to read different works on logic, 
and finds them to vary exceedingly in the mode of treat- 
ment, or at least of expression. 

If, when reduced to language, there be three parts of 
logic, terms, propositions, and syllogisms, there must be 
as many different kinds of thought or operations of mind. 
These are usually called — 

T. Simple apprehension. 

2. Judgment. 

3. Reasoning or discourse. 

The first of these, Simple Apprehension, is the act of 
mind by which we merely become aware of something, 
or have a notion, idea, or impression of it brought into 
the mind. The adjective simple means apart from other 
things, and apprehension the taking hold by the mind. 
Thus the name or term Iron instantaneously makes the 



12 THE THREE PARTS OF [less. 

mind think of a strong and very useful metal, but does 
not tell us anything about it, or compare it with any thing 
else. The words sun^ Jtipiter^ Sirius^ St PauVs Cathe- 
dral^ are also terms which call up into the mind certain 
well-known objects, which dwell in our recollection even 
when they are not present to our senses. In fact, the use 
of a term, such as those given as examples, is merely as a 
substitute for the exhibition of the actual things named. 

Judgment is a different action of mind, and consists in 
comparing together two notions or ideas of objects de- 
rived from simple apprehension, so as to ascertain whe- 
ther they agree or differ. It is evident, therefore, that we 
cannot judge or compare unless we are conscious of two 
things or have the notions of two things in the mind at 
the same time. Thus if I compare Jupiter and Sirius I 
first simply apprehend each of them ; but bringing them 
into comparison I observe that they agree in being small, 
bright, shining bodies, which rise and set and move 
round the heavens with apparently equal speed. By 
minute examination, however, I notice that Sirius gives 
a twinkling or intermittent light, whereas Jupiter shines 
steadily. More prolonged observation shews that Ju- 
piter and Sirius do not really move with equal and 
regular speed, but that the former changes its position 
upon the heavens from night to night in no very simple 
manner. If the comparison be extended to others of the 
heavenly bodies which are apprehended or seen at the 
same time, I shall find that there are a multitude of stars 
which agree with Sirius in giving a twinkling light and 
in remaining perfectly fixed in relative position to each 
other, whereas two or three other bodies may be seen 
which resemble Jupiter in giving a steady light, and also 
in changing their place from night to night among the 
fixed stars. I have now by the action of judgment 
formed in my mind the general notion of fixed starsy by 



IL] LOGICAL DOCTRINE. 13 

bringing together mentally a number of objects which 
agree ; while from several other objects I have formed the 
general xvolion oi plunets. Comparing the two general 
notions together, I find that they do not possess the same 
qualities or appearances, which I state in the proposition, 
** Planets are not fixed stars." 

I have introduced the expression "General Notion" as 
if the reader were fully acquainted with it. But though 
philosophers have for more than two thousand years con- 
stantly used the expressions, general notion, idea, con- 
ception, concept, &c., they have never succeeded in 
agreeing exactly as to the meaning of the terms. One 
class of philosophers called Nominalists say that it is all a 
matter of names, and that when we join together Jupiter, 
Mars, Saturn, Venus, &c., and call them planets^ the 
common name is the bond between them in our minds. 
Others, called Realists, have asserted that besides these 
particular planets there really is something which com- 
bines the properties common to them all without any of 
the differences of size, colour, or motion which distin- 
guish them. Every one allows in the present day how- 
ever that nothing can physically exist corresponding to a 
general notion, because it must exist here or there, of this 
size or of that size, and therefore it would be one particu- 
lar planet, and not any planet whatever. The Nominal- 
ists, too, seem equally wrong, because language, to be of 
any use, must denote something, and must correspond, as 
we have seen, to acts of mind. If then proper names 
raise up in our minds the images of particular things, like 
the sun, Jupiter, &c., general names should raise up 
general notions. 

The true opinion seems to be that of the philoso- 
phers called Conceptualists, who say that the general no- 
tion is the knowledge in the mind of the common pro- 
perties or resemblances of the things embraced undef 



14 THE THREE PARTS OF [less, 

the notion. Thus the notion planet really means the 
consciousness in anybody's mind that there are certain 
heavenly bodies which agree in giving a steady light 
and in moving about the heavens differently from the 
fixed stars. It should be added, however, that there are 
many, including Sir W. Hamilton, who would be counted 
as Nominalists and who yet hold that with the general 
name is associated a consciousness of the resemblance 
existing between the things denoted by it. Between this 
form of the doctrine and conceptualism it is not easy to 
draw a precise distinction, and the subject is of too de- 
batable a character to be pursued in this work. 

It will appear in the course of these lessons that the 
whole of logic and the whole of any science consists in so 
arranging the individual things we meet in general no- 
tions or classes, and in giving them appropriate general 
names or terms, that our knowledge of them may be 
made as simple and general as possible. Every general 
notion that is properly formed admits of the statement of 
general laws or truths; thus of the planets we may affirm 
that they move in elliptic orbits round the sun from west 
to east ; that they shine with the reflected light of the 
sun; and so on. Of the fixed stars we may affirm that 
they shine with their own proper light; that they are 
incomparably more distant than the planets ; and so on. 
The whole of reasoning will be found to arise from this 
faculty of judgment, which enables us to discover and 
affirm that a large number of objects have similar pro- 
perties, so that whatever is known of some may be in- 
ferred and asserted of others. 

It is in the application of such knowledge that we 
employ the third act of mind called discourse or reason- 
ing, by which from certain judgments we are enabled, 
without any new reference to the real objects, to form a 
new judgment. If we know that iron comes under the 



11.] LOGICAL DOCTRINE. 15 

general notion of metal, and that this notion comes under 
the still wider notion of element, then without further 
examination of iron we know that it is a simple unde- 
composable substance called by chemists an element. Or 
if from one source of information we learn that Neptune 
is a planet, and from another that planets move in ellip- 
tic orbits, we can join these two portions of knowledge 
together in the mind, so as to elicit the truth that Nep- 
tune moves in an elliptic orbit. 

Reasoning or Discourse, then, may be defined as the 
progress of the mind from one or more given propositions 
to a proposition different from those given. Those pro- 
positions from which we argue are called Premises, and 
that which is drawn from them is called the Conclusion, 
The latter is said to follow, to be concluded, inferred or col- 
lected from them ; and the premises are so called because 
they are put forward or at the beginning (Latin prcB^ be- 
fore, and mitto, I send or put). The essence of the pro- 
cess consists in gathering the truth that is contained in 
the premises when joined together, and carrying it with 
us into the conclusion, where it is embodied in a new 
proposition or assertion. We extract out of the pre- 
mises all the information which is useful for the purpose 
in view — and this is the whole which reasoning accom- 
plishes. 

I have now pointed out the three parts of logical doc- 
trine. Terms, Propositions, and Reasoning or Syllogism, 
into which the subject is conveniently divided. To the 
consideration of these parts we shall proceed. But it 
may be mentioned that a fourth part has often been 
added called Method, which is concerned with the ar* 
rangement of the parts of any composition. 

It is sometimes said that what proposition is to term, 
and what syllogism is to proposition, such is method to 
syllogism, and that a fourth division is necessary to com- 



i6 TERMS, AND THEIR _ [LES& 

plete the doctrine of Logic. It is at any rate certain 
however that this fourth part is much inferior in import- 
ance and distinctness to the preceding three ; and all that 
will be said of it is to be found in Lesson xxiv. 



LESSON III. 

TERMS, AND THEIR VARIOUS KINDS, 

It has been explained in the preceding lesson that every 
assertion or statement expresses the agreement or dif- 
ference of two things, or of two general notions. In 
putting the assertion or statement into words, we must 
accordingly have words suitable for drawing the attention 
of the mind to the things which are compared, as well as 
words indicating the result of the comparison, that is to 
say, the fact whether they agree or differ. The words by 
which we point out the things op classes of things in 
question are called Terms, and the words denoting the 
comparison are said to form the Copula. Hence a com- 
plete assertion or statement consists of two terms and a 
copula, and when thus expressed it forms a Propositioii. 
Thus in the proposition " Dictionaries are useful books," 
the two terms are dictionaries and useftil books; the co- 
pula is the verb are, and expresses a certain agreement of 
the class dictionaries with the class of useful books con- 
sisting in the fact that the class of dictionaries forms part 
of the class of useful books. In this case each tern) con- 
sists of only one or two words, but any number of words 
may be required to describe the notions or classes com- 



III.] VARIOUS KINDS. 17 

pared together. In the proposition "the angles at the 
base of an isosceles triangle are equal to each other," the 
first term requires nine words for its expression, and the 
second term, four words (equal to each other); and there 
is no limit to the number of words which may be em- 
ployed in the formation of a term. 

A term is so called because it forms one end (Latin, 
terminus) of a proposition, and strictly speaking it is a 
tenn only so long as it stands in the proposition. But 
we commonly speak of a term or a name meaning any 
noun, substantive or adjective, or any combination of 
words denoting an object of thought, whether that be, as 
we shall shortly see, an individual thing, a group of things, 
a quality of things, or a group of qualities. It would be 
impossible to define a name or term better than has been 
done by Hobbes : "A name is a word taken at pleasure 
to serve for a mark, which may raise in our mind a 
thought like to some thought which we had before, and 
which, being pronounced to others, may be to them a 
sign of what thought the speaker had before in his mind." 

Though every term or name consists of words it is 
not every word which can form a name by itself. We 
cannot properly say " Not is agreeable" or " Probably is 
not true ;" nothing can be asserted of a preposition, an 
adverb, and certain other parts of speech, except indeed 
that they are prepositions, adverbs, &c. No part of 
speech except a noun substantive, or a group of words 
used as a noun substantive, can form the subject or first 
term of a proposition, and nothing but a noun substan- 
tive, an adjective, the equivalent of an adjective, or a 
verb, can form the second term or predicate of a propo- 
sition. It may indeed be questioned whether an adjec- 
tive can ever form a term alone ; thus in " Dictionaries 
are useful," it may be said that the substantive things or 
books is understood in the predicate, the complete sen- 

2 



l8 TERMS, AND THEIR [less, 

tence being " Dictionaries are useful books f but as this 
is a disputed point we will assume that words are divided 
into two kinds in the following manner : — 

Words which stand, or appear to stand alone as com- 
plete terms, namely the substantive and adjective, and 
certain parts of a verb, are called categorematic words, 
from the Greek word KaTT)yop€(o, to assert or predicate. 

Those parts of speech, on the other hand, such as 
prepositions, adverbs, conjunctions, &c., which can only 
form parts of names or terms are called syncategorematio 
words, because they must be used wz^A other words in 
order to compose terms (Greek o-vv, with, and Kan^yopeca). 
Of syncategorematic words we need not take further 
notice except so far as they form part of categorematic 
terms. 

We have now to consider the various kinds and pecu- 
liarities of terms, so as to gain a clear idea of what they 
mean. Terms are first of all distinguished into si7igular 
or individual, and general or common terms, this being a 
very obvious division, but one of much importance. A 
Singular term is one which can denote only a single ob- 
ject, so long at least as it is used in exactly the same 
meaning ; thus the Emperor of the French, the Atlantic 
Ocean, St Paul's, William Shakspeare, the most pre- 
cious of the metals, are singular terms. All proper names 
belong to this class ; for though John Jones is the name 
of many men, yet it is used not as meaning any of these 
men, but some single man — it has, in short, a different 
meaning in each case, just as London, the name of our 
capital, has no connexion in meaning with London m 
Canada. 

General terms, on the contrary, are applicable in the 
same sense equally to any 07te of an indefinite number of 
objects which resemble each other in certain qualities. 
Thus metal is a general name because it may be applied 



III.] VARIOUS KINDS, 19 

mdififerently to gold, silver, copper, tin, aluminium, or any 
of about fifty known substances. It is not the name of 
any one of these more than any other, and it is in fact 
applied to any substance which possesses metallic lustre, 
which cannot be decomposed, and which has certain 
other qualities easily recognised by chemists. Nor is the 
number of substances in the class restricted; for as new 
kinds of metal are from time to time discovered they are 
added to the class. Again, while Mars, Jupiter, Saturn, 
&c., are singular terms, since each can denote only a 
single planet, the term planet is a general one, being 
applicable to as many bodies as may be discovered to 
revolve round the sun as the earth does. 

We must carefully avoid any confusion between ge- 
neral and collective terms. By a collective term we 
mean the name of a number of things when ail joined 
together as one whole ; like the soldiers of a regiment, 
the men of a jury, the crew of a vessel : thus a collective 
term is the name of all, but not of each. A general term, 
on the other hand, is the name of a number of things, 
but of each of them separately, or, to use the technical 
expression, distributively. Soldier, juryman, sailor, are 
the general names which may belong to John Jones, 
Thomas Brown, &c., but we cannot say that John Jones 
is a regiment, Thomas Brown a jury, and so on. The 
distinction is exceedingly obvious when thus pointed out, 
but it may present itself in more obscure forms, and is 
then likely to produce erroneous reasoning, as will be 
pointed out in Lesson XX. It is easy to see that we must 
not divide terms into those which are general and those 
which are collective, because it will often happen that 
the same term is both general and collective, according 
as it is regarded. Thus, library is collective as regards 
the books in it, but is general as regards the great num- 
ber of different libraries, private or public, which exist 

2 — 2 



20 TERMS, AND THEIR 

Regiment is a collective term as regards the soldiers 
which compose it, but general as regards the hundred 
different regiments, the Coldstream Guards, the High- 
land regiment, the Welsh Fusiliers, and the rest, which 
compose the British standing army. Army, again, is a 
collective whole, as being composed of a number of regi- 
ments organized together. Year is collective as regards 
the months, weeks, or days of which it consists, but is 
general as being the name either of 1869 or 1870, or any 
period marked by a revolution of the earth round the sun. 

We have not always in the English language suffi- 
cient means of distinguishing conveniently between the 
general and collective use of terms. In Latin this dis- 
tinctive use was exactly expressed by omnes^ meaning all 
distributively, and cuncti meaning all taken together, a 
contracted form of conjuncti (joined together). In English 
all me7i may mean a7iy man or all men together. Even 
the more exact word every is sometimes misused, as in 
the old proverb, ' Every little makes a mickle,' where it is 
obvious that every little portion cannot by itself make 
much, but only when joined to other little portions. 

A second important distinction between terms is that 
of concrete terms and abstract terms ; and it cannot be 
better described than in the words of Mr Mill, by saying 
that a concrete name is the name of a thing, the abstract 
name is the name of a quality, attribute, or circumstance 
of a thing. Thus red house is the name of a physically- 
existing thing, and is concrete; redness is the name of 
one quality of the house, and is abstract. The word 
abstract means drawn from (Latin, abstractus, from ads- 
trahere^ to draw away from), and indicates that the quality 
redness is thought of in the mind apart from all the other 
qualities which belong to the red house, or other red 
object. But though we can think of a quality by itself, 
we cannot suppose that the quality can exist physically 



III.] VARIOU:^ KINDS, 21 

apart from the matter in which it is manifest to us. Red- 
ness means either a notion in the mind, or it means that 
in red objects which excites the notion. 

The reader should carefully observe that adjectives 
are concrete, not abstract. If we say that a book is use- 
ful, it is to the book we apply the adjective useful, and 
usefulness is the abstract noun which denotes the quality; 
similarly, the adjectives equal, grateful, reverent, ratio- 
nal, are the names of things, and the corresponding abs- 
tract nouns are equality, g7'atitude, reve7'ence, rationality. 
This distinction will become more apparent in reading 
Lesson V. 

It is a good exercise to try and discover pairs of cor- 
responding concrete and abstract names ; thus animal 
has animality ; miser, miserliness ; old, agedness, or old 
age ; substance, substantiality ; soap, soapiness ; shrub, 
shrubbiness ; and so on. But it by no means follows that 
an abstract word exists for each concrete ; table hardly has 
an abstract tabularity ; and though ink has inkiness, we 
should not find the abstract of pen. It is by the accidents 
of the history of language that we do or do not possess 
abstract names ; and there is a constant tendency to in- 
vent new abstract words in the progress of time and 
science. 

Unfortunately concrete and abstract names are fre- 
quently confused, and it is by no means always easy to 
distinguish the meanings. Thus relation properly is the 
abstract name for the position of two people or things to 
each other, and those people are properly called relatives 
(Latin, relativus, one who is related). But we constantly 
speak now of relations, meaning the persons themselves ; 
and when we want to indicate the abstract relation 
they have to each other we have to invent a new abstract 
name relationship. Nation has long been a concrete 
term, though from its form it was probably abstract at 



22 TERMS, AND THEIR [less. 

first ; but so far does the abuse of language now go, 
especially in newspaper writing, that we hear of a nation- 
ality meaning a nation, although of course if nation is 
the concrete, nationality ought to be the abstract, mean- 
ing the quality of being a nation. Similarly, action^ 
intention, extension, conception, and a multitude of other 
properly abstract names, are used confusedly for the corre- 
sponding concrete, namely, act, intent, extent, concept, &c. 
Production is properly the condition or state of a person 
who is producing or drawing something forth ; but it has 
now become confused with that which is produced, so 
that we constantly talk of the productions of a country, 
meaning the products. The logical terms, Proposition, 
Deduction, Induction, Syllogism, are all properly abstract 
words, but are used concretely for a Proposition, a De- 
duction, an Induction, a Syllogism ; and it must be al- 
lowed that logicians are nearly as bad as other people in 
confusing abstract and concrete terms. Much injury is 
done to language by this abuse. 

Another very obvious division of terms is between 
those which are positive, and those which are negative. 
The difference is usually described by saying that posi- 
tive terms signify the existence or possession of a quality, 
as in grateful, metallic, organic, etc., while the correspond- 
ing negatives signify the absence of the same quahties 
as in ungrateful, non-metallic, inorganic. The negative 
terms may be adjectives as above, or substantives, con- 
crete or abstract ; thus ingratitude, inequality, incon- 
venience are abstract negative terms; and individuals, 
unequals, &c. are concrete negatives. We usually consider 
as negative terms any which have a negative prefix such 
as not, non, un, in, &c. ; but there are a great many terms 
which serve as negatives without possessing any mark of 
their negative character. Darkness is the ne^^ative of 
light or lightness, since it means the absence of light; 



III.] VARIOUS KINDS. 23 

compound is the negative of element, since we should 
give the name of compound to whatever can be decom- 
fosed^ and element is what cannot be decomposed ; theo- 
retically speaking every term has its corresponding nega- 
tive, but it by no means follows that language furnishes 
the term ready-made. Thus table has the corresponding 
adjective tabular, but there is no similar negative untabu- 
larj one man may be called a bookworm, but there is no 
negative for those who are not bookworms, because no 
need of the expression has been felt. A constant process 
of invention of new negative terms goes on more rapidly 
perhaps than is desirable, for when an idea is not often 
referred to it is better to express it by a phrase than add 
to the length of the dictionary by a new-created word. 

It would seem that in many cases a negative term 
implies the presence of some distinct quality or fact. 
Thus inconvenience doubtless implies the absence of 
cofivenience^ but also the presence of positive trouble or 
pain occasioned thereby. Unhappiness is a negative 
term, but precisely the same notion is expressed by the 
positive term misery. The negative of healthy is un- 
healthy, but the positive term sickly serves equally well. 
It thus appears to be more a matter of accident than 
anything else whether a positive or negative term is used 
to express any particular notion. All that we can really 
say is that every positive term necessarily implies the 
existence of a corresponding negative term, which maybe 
the name of all those things to which the positive name 
cannot be applied. Whether this term has been invented 
or not is an accident of language: its existence may be 
assumed in logic. 

The reader may be cautioned against supposing that 
every term appearing to be of a negative character on 
account of possessing a negative prefix is really so. The 
participle unloosed certainly appears to be the negative of 



24 TERMS, AND THEIR [less. 

loosed; Dut the two words mean exactly the same thing, 
the prefix un not being really the negative ; invahiable^ 
again, means not what is devoid of value, but what is so 
valuable that the value cannot be measured; and a 
shameless action can equally be called by the positive 
term, a shameful action. Other instances might no 
doubt be found. 

Great care should be taken to avoid confusing terms 
which express the presence or absence of a quality with 
those which describe its degree. Less is not the negative 
oi greater because there is a third alternative, equal. The 
true negative oi greater \s^ not-greater^ and this is equiva- 
lent to either equal or less. So it may be said that dis- 
agreeable is not the simple negative of agreeable^ because 
there may be things which are neither one nor the other, 
but are i7idifferent to us. It would not be easy to say 
offhand whether every action which is not honest is dis- 
honest, or whether there may not be actions of an inter- 
mediate character. The rule is that wherever the question 
is one of degree or quantity a medium is possible, and 
the subject belongs rather to the science of quantity 
than to simple logic ; where the question is one of the 
presence or absence of a quality, there cannot be more 
than two alternatives, according to one of the Primary 
Laws of Thought, which we will consider in Lesson XIV. 
In the case of quantity we may call the extreme terms 
opposites ; thus less is the opposite of greater, disagreeable 
of agreeable ; in the case of mere negation we may call 
the terms negatives or contradictories, and it is really 
indifferent in a logical point of view which of a pair of 
contradictory terms we regard as the positive and which 
as the negative. Each is the negative of the other. 

Logicians have distinguished from simple negative 
terms a class of terms called privative, such as bli7id^ 
dead, &:c. Such terms express tliat a thing has been 



III. J VARIOUS KINDS. 25 

deprived of a quality which it before possessed, or was 
capable of possessing, or usually does possess. A man 
may be born blind, so that he never did see, but he pos- 
sesses the organs which would have enabled him to see 
except for some accident. A stone or a tree could not 
have had the faculty of seeing under any circumstances. 
No mineral substance can properly be said to die or to 
be dead, because it was incapable of life ; but it may be 
called uncrystallized because it might have been in the 
form of a crystal. Hence we apply a privative term to 
anything which has not a quality which it was capable of 
having ; we apply a negative term to anything which has 
not and could not have the quality. It is doubtful however 
whether this distinction can be properly carried out, and 
it is not o^ very much importance. 

It IS further usual to divide terms according as they 
are relative or absolute, that is, non-relative. The adjective 
absolute means whatever is ''loosed from connection 
with anything else" (Latin ab^ from, and solutus^ loosed); 
whereas relative means that which is carried in thought, 
at least, into connection with something else. Hence a 
relative term denotes an object which cannot be thought 
of without reference to some other object, or as part of a 
larger whole. A father cannot be thought of but in rela- 
tion to a child, a monarch in relation to a subject, a shep- 
herd in relation to a flock ; thus father, monarch, and 
shepherd are relative terms, while child, subject, and 
flock are the correlatives (Latin con^ with, and relativMs)^ 
or those objects which are necessarily joined in thought 
with the original objects. The very meaning, in fact, of 
father is that he has a child, of monarch that he has 
subjects, and of shepherd that he has a flock. As ex- 
amples of terms which have no apparent relation to any- 
thing else, I may mention water, gas, tree. There does 
not seem to me to be anything so habitually associated 



26 TERMS, AND THEIR [LESS. 

with water that we must think of it as part of the same 
idea, and gas, tree, and a multitude of other terms, also 
denote objects which have no remarkable or permanent 
relations such as would entitle the terms to be called rela- 
tives. They may therefore be considered absolute or 
non-relative terms. 

The fact, however, is that everything must really have 
relations to something else, the water to the elements of 
which it is composed, the gas to the coal from which it is 
manufactured, the tree to the soil in which it is rooted. 
By the very laws of thought, again, no thing or class of 
things can be thought of but by separating them from 
other existing things from which they differ. I cannot use 
the term mortal without at once separating all existing 
or conceivable things into the two groups mortal and 
un?nortal; metal, element, organic substance, and every 
other term that could be mentioned, would necessarily 
imply the existence of a correlative negative term, non- 
metallic, compound, inorganic substance, and in this 
respect therefore every term is undoubtedly relative. 
Logicians, however, have been content to consider as 
relative terms those only which imply some peculiar and 
striking kind of relation arising from position in time or 
space, from connexion of cause and effect, &c. ; and it 
is in this special sense therefore the student must use the 
distinction. 

The most important varieties of terms having been 
explained, it is desirable that the reader should acquire a 
complete familiarity with them by employing the exercises 
at the end of the book. The reader is to determine con- 
cerning each of the terms there given : — 

1. Whether it is a categorematic or syncategore- 

matic term. 

2. Whether it is a general or a singular term. 

3. Whether it is collective or distributive. 



III.] VARIOUS KINDS, 27 

4. Whether it is concrete or abstract. 

5. Whether it is positive, or negative, or privative. 

6. Whether it is relative or absolute. 

It will be fully pointed out in the next lesson that 
most terms have more than one meaning; and as the one 
meaning may be general and the other singular, the one 
concrete and the other abstract, and so on, it is absolute- 
ly necessary that the reader should first of all choose 
one precise meaning of the term which he is examining. 
And in answering the questions proposed it is desirable 
he should specify the way in which he regards it. Taking 
the word sovereign, we may first select the meaning in 
which it is equivalent to monarch ; this is a general term 
in so far as it is the name of any one of many monarchs 
living or dead, but it is singular as regards the inhabit- 
ants of any one country. It is clearly categorematic, 
concrete, and positive, and obviously relative to the sub- 
jects of the monarch. 

Read Mr MilFs chapter on Names y System 0/ Logic 
Book I. chap. 2. 



LESSON IV. 

OF THE AMBIGUITY OF TERMS. 

There is no part of Logic which is more really useful 
than that v/hich treats of the ambiguity of terms, that is 
of the uncertainty and variety of meanings belonging to 
words. Nothing indeed can be of more importance to 
the attainment of correct habits of thinking and reason- 
ing than a thorough acquaintance with the great imper- 
fections of language. Comparatively few terms have one 



28 OF THE AMBIGUITY [less. 

sinjdjle clear meaning and one meaning only, and when- 
ever two or more meanings are unconsciously confused 
together, we inevitably commit a logical fallacy. If, for 
instance, a person should argue that " punishment is an 
evil," and according to the principles of morality "no 
evil is to be allowed even with the purpose of doing 
good," we might not at the first moment see how to avoid 
the conclusion that " no punishments should be allowed," 
because they cause evil. A little reflection will show that 
the word evil is here used in two totally different senses ; 
in the first case it means physical evil or pain ; in the 
second moral evil, and because moral evil is never to be 
committed, it does not follow that physical evils are never 
to be inflicted, for they are often the very means of pre- 
venting moral evil. 

Another very plausible fallacy which has often been 
put forth in various forms is as follows : " A thoroughly 
benevolent man cannot possibly refuse to relieve the poor, 
and since a person who cannot possibly act otherwise 
than he does can claim no merit for his actions, it follows 
that a thoroughly benevolent man can claim no merit for 
his actions." According to this kind of argument a man 
would have less merit in proportion as he was more 
virtuous, so as to feel greater and greater difficulty in 
acting wrongly. That the conclusion is fallacious every 
one must feel certain, but the cause of the fallacy can 
only be detected by observing that the words ca7tnot 
possibly have a double meaning, in the first case referring 
to the influence of moral motives or good character, and 
in the second to circumstances entirely beyond a person's 
control ; as, for instance, the compulsion of the laws, the 
want of money, the absence of personal liberty. The 
more a person studies the subtle variations in the mean- 
ing of common words, the more he will be convinced of 
the dangerous nature of the tools he has to use in all 



IV.] OF TERMS. 29 

communications and • arguments. Hence I must ask 
much attention to the contents of this Lesson. 

Terms are said to be univocal when they can suggest 
to the mind no more than one single definite meaning. 
They are called equivocal or ambiguous when they have 
two or more different meanings. It will be observed, 
however, that a term is not equivocal because it can be 
apphed to many objects when it is applied in the same 
sense or meaning to those different objects. Thus cathe- 
dral is the name of St Paul's, the York Minster, and the 
principal churches of Salisbury, Wells, Lincoln and a 
number of other cities, but it is not ambiguous, because 
all these are only various instances of the same meaning ; 
they are all objects of the same description or kind. 
The word cathedral is probably univocal or of one logical 
meaning only. The word church, on the other hand, is 

.equivocal, because it sometimes means the building in 
which religious worship is performed, sometimes the body 
of persons who belong to one sect or persuasion, and 
assemble in churches. Sometimes also the church 
means the body of the clergy as distinguished from the 
laity; hence there is a clear difference in the sense or 
meaning with which the word is used at different times. 

Instances of univocal terms are to be found chiefly in 
technical and scientific language. Steam-engine, gas- 
ometer, railway train, permanent way, and multitudes of 

^such technical names denoting distinct common objects, 
are sufficiently univocal. In common life the names 
penny, mantelpiece, teacup, bread and butter, have a suf- 
ficiently definite and single meaning. So also in chemistry, 
oxygen, hydrogen, sulphate of copper, alumina, lithia, 
and thousands of other terms, are very precise, the words 
themselves having often been invented in very recent 
years, and the meanings exactly fixed and maintained 
invariable. Every science has or ought to have a series 



30 OF THE AMBIGUITY [LESS. 

of terms equally precise and certain in meaning. (See 
Lesson xxxiil.) The names of individual objects, build- 
ings, events, or persons, again, are usually quite certain 
and clear, as in Julius Caesar, William the Conqueror, the 
first Napoleon, Saint Peter's, Westminster Abbey, the 
Great Exhibition of 185 1, and so on. 

But however numerous may be the univocal terms 
which can be adduced, still the equivocal terms are asto- 
nishingly common. They include most of the nouns and 
adjectives which are in habitual use in the ordinary 
intercourse of life. They are called ambiguous from the 
Latin verb ambigo^ to wander, hesitate, or be in doubt; or 
again hoinonyrnous^ from the Greek oyt-os, like, and ovofia, 
name. Whenever a person uses equivocal words in such 
a way as to confuse the different meanings and fall into 
error, he may be said to commit the fallacy of Equivoca- 
tion in the logical meaning of the name (see Lesson XX.) ; 
but in common life a person is not said to equivocate 
unless he uses words consciously and deceitfully in a 
manner calculated to produce a confusion of the true and 
apparent meanings. 

I will now describe the various kinds and causes of 
ambiguity of words, following to some extent the inter- 
esting chapters on the subject in Dr Watts' Logic, In 
the first place we may distinguish three classes of equi- 
vocal words, according as they are — 

1. Equivocal in sound only. 

2. Equivocal in spelling only. 

3. Equivocal both in sound and spelling. 

The first two classes are comparatively speaking of very 
slight importance, and do not often give rise to serious 
error. They produce what we should call trivial mis- 
takes. Thus we may confuse, when spoken only, the 
words right, wright and rite (ceremony) ; also the words 
rein, rain and reign, might and mite, &:c. Owing partly 



IV.] OF TERMS. 31 

to defects of pronunciation mistakes are not unknown 
between the four words air^ hair, hare and heir. 

Words equivocal in spelling but not in sound are such 
as tear (a drop), and tear pronounced tare, meaning a 
rent in cloth ; or lead, the mefal, and lead, as in follow- 
ing the lead of another person. As little more than mo- 
mentary misapprehension, however, can arise from such 
resemblance of words, we shall pass at once to the class 
of words equivocal both in sound and spelling. These I 
shall separate into three groups according as the equivo- 
cation arises — 

1. From the accidental confusion of different words. 

2. From the transfer of meaning by the association of 

ideas. 

3. From the logical transfer of meaning to analogous 

objects. 

I. Under the first class we place a certain number 
of curious but hardly important cases in which ambi- 
guity has arisen from the confusion of entirely different 
words, derived from different languages or from differ- 
ent roots of the same language, but which have in 
the course of time assumed the same sound and spell- 
ing. Thus the word mean denotes either that which 
is medium or mediocre, from the French moyen and 
the Latin medius, connected with the Anglo-Saxon 
mid, or middle; or it denotes what is low-minded and 
base, being then derived from the Anglo-Saxon Gemoene, 
which means " that belonging to the moene or many,'* 
whatever in short is vulgar. The verb to 7neaTi can 
hardly be confused with the adjective mean, but it comes 
from a third distinct root, probably connected with the 
Sanscrit verb, to think. 

As other instances of this casual ambiguity, I may 
mention rent, a money payment, from the French reiite 
{/endre, to return), or a tear, the result of the action of 



32 OF THE AMBIGUITY [less. 

rending^ this word being of Anglo-Saxon origin and one 
of the numerous class beginning in r or wr, which imitate 
more or less perfectly the sound of the action which they 
denote. Pounds from the \.2X\n pondtis^ a weight, is con- 
fused with poundy in the sense of a village pinfold for 
cattle, derived from the Saxon pyndan, to pen up. Fellj 
a mountain, is a perfectly distinct word from fell, a skin 
or hide ; and pulse, a throb or beating, and pulse, peas, 
beans, or potage, though both derived from the Greek or 
Latin, are probably quite unconnected words. It is 
curious that gin, in the meaning of trap or machine, is a 
contracted form of engine, and when denoting the spirit- 
uous liquor is a corruption of Geneva, the place where the 
spirit was first made. 

Certain important cases of confusion have been de- 
tected in grammar, as between the numeral one, derived 
from an Aryan root, through the Latin unus, and the in- 
determinate "pronoun,* one (as in ^' one ought to do one's 
duty "), which is really a corrupt form of the French 
word ho7nme or man. The Germans to the present day 
use man in this sense, as in man sagt, i.e, one says. 

2. By far the largest part of equivocal words have 
become so by a transfer of tlie meaning from the thing 
originally denoted by the word to some other thing 
habitually connected with it so as to become closely as- 
sociated in thought. Thus, in Parliamentary language, 
the House means either the chamber in which the mem- 
bers meet, or it means the body of members who happen 
to be assembled in it at any time. Similarly, the word 
church originally denoted the building (KvptaKov, the 
Lord's House) in which any religious worshippers assem- 
ble, but it has thence derived a variety of meanings ; it 
may mean a particular body of worshipper's accustomed 
to assemble in any one place, in which sense it is used in 
Acts xiv. 23; or it means any body of persons holding 



IV.] OF TERMS. 33 

the same opinions and connected in one organization, as 
in the Anglican, or Greek, or Roman Catholic Church ; 
it is also sometimes used so as to include the laity as well 
as the clergy ; but more generally perhaps the clergy and 
religious authorities of any sect or country are so strongly 
associated with the act of worship as to be often called 
the church par excellence. It is quite evident moreover 
that the word entirely differs in meaning according as it 
is used by a member of the Anglican, Greeh; Roman 
Catholic, Scotch Presbyterian, or any other existing 
church. 

The word foot has suffered several curious but very 
evident transfers of meaning. Originally it denoted the 
foot of a man or an animal, and is probably connected in 
a remote manner with the Latin pes^ pedis^ and the Greek 
trov?, TToSo? ; but since the length of the foot is naturally 
employed as a rude measure of length, it came to be 
applied to a fixed measure of length ; and as the foot is 
at the bottom of the body the name was extended by 
analogy to the foot of a mountain, or the feet of a table ; 
by a further extension, any position, plan, reason, or 
argument on which we place ourselves and rely, is called 
the foot or footing. The same word also denotes soldiers, 
who fight upon their feet, or infantry, and the measured 
part of a verse having a definite length. That these very 
different meanings are naturally connected with the ori- 
ginal meaning is evident from the fact that the Latin 
and Greek words for foot are subject to exactly similar 
series of ambiguities. 

It would be a long task to trace out completely the 
various and often contradictory meanings of the word 
fellow. Originally a fellow was what yc'//t?7e/i' another, that 
is a companion ; thus it came to mean the other of a pair, 
as one shoe is the fellow of the other, or simply an equal, 
as when we say that Shakspeare "hath not a fellow'' 



34 OF THE AMBIGUITY [less. 

From the simple meaning of companion again it comes 
t6 denote vaguely a person, as in the question "What 
fellow is that?" but then there is a curious confusion of 
depreciatory and endearing power in the word ; when a 
man is called a 77iere fellow^ or simply a fellow in a par- 
ticular tone of voice, the name is one of severe contempt ; 
alter the tone of voice oi the connected words in the least 
degree, and it becomes one of the most sweet and en- 
dearing appellations, as when we speak of a dear or 
good fellow. We may still add the technical meanings of 
the name as applied in the case of a Fellow of a College, 
or of a learned society. 

Another good instance of the growth of a number of 
different meanings from a single root is found in the 
word post. Originally a post was something posited^ or 
placed firmly in the ground, such as an upright piece of 
wood or stone ; such meaning still remains in the cases 
of a lamp-post, a gate-post, signal-post, &c. As a post 
would often be used to mark a fixed spot of ground, as in 
a mile-post, it came to mean the fixed or appointed place 
where the post was placed, as in a military post, the post 
of danger or honour, &c. The fixed places where horses 
were kept in readiness to facilitate rapid travelling during 
the times of the Roman empire were thus called posts, 
and thence the whole system of arrangement for the con- 
veyance of persons or news came to be called the posts. 
The name has retained an exactly similar meaning to the 
present day in most parts of Europe, and we still use it 
in post-chaise, post-boy, post-horse and postillion. A 
system of post conveyance for letters having been organ- 
ised for about two centuries in England and other coun- 
tries, this is perhaps the meaning most closely associated 
with the word post at present, and a number of expres- 
sions have thus arisen, such as post-office, postage, postal- 
guide, postman, postmaster, postal-telegraph, &c. Curi* 



IV.] OF TERMS, 35 

ously enough we now have iron letter-posts, in which the 
word post is restored exactly to its original meaning. 

Although the words described above were selected on 
account of the curious variety of their meanings, I do not 
hesitate to assert that che majority of common nouns 
possess various meanings in greater or less number. Dr 
Watts, in his Logic, suggests that the words book, bible, 
fish, house, and elephant, are univocal terms, but the 
reader would easily detect ambiguities in each of them. 
Thus fish bears a very different meaning in natural his- 
tory from what it does in the mouths of unscientific per- 
sons, who include under it not only true fishes, but shell- 
fish or mollusca, and the cetacea, such as whales and 
seals, in short all swimming animals, whether they have 
the character of true fish or not. Elephant, in a station- 
er's or bookseller's shop, means a large kind of paper 
instead of a large animal. Bible sometimes means any 
particular copy of the Bible, sometimes the collection 
of works constituting the Holy Scriptures. The word 
man is singularly ambiguous ; sometimes it denotes man 
as distinguished from woman ; at other times it is cer- 
tainly used to include both sexes ; and in certain recent 
election cases lawyers were unable to decide whether the 
word man as used in the Reform Act of 1867 ought or 
ought not to be interpreted so as to include women. On 
other occasions 7na7i is used to denote an adult male as 
distinguished from a boy, and it also often denotes one 
who is emphatically a man as possessing a masculine 
character. Occasionally it is used in the same way as 
groom, for a servant, as in the proverb, " Like master, 
like man." At other times it stands specially for a hus' 
band. 

3. Among ambiguous words we must thirdly distinguish 
those which derive their various meanings in a somewhat 
different manner, namely by analogy or real resemblance 

3—2 



S6 THE AMBIGUITY OF TERMS, [less. IV. 

When we speak of a sweet taste, a sweet flower, a sweet 
tune, a sweet landscape, a sweet face, a sweet poem, it is 
evident that we apply one and the same word to very 
different things ; such a concrete thing as lump-sugar can 
hardly be compared directly with such an intellectual 
existence as Tennyson's May Queen. Nevertheless if the 
word sweet is to be considered ambiguous, it is in a dif- 
ferent way from those we have before considered, because 
all the things are called sweet on account of a peculiar 
pleasure which they yield, which cannot be described 
otherwise than by comparison with sugar. In a similar 
way, we describe a pain as sharp, a disappointment as 
bitter, a person's temper as sour, the future as bright or 
gloomy, an achievement as brilliant ; all these adjectives 
implying comparison with bodily sensations of the sim- 
plest kind. The adjective br'illiant is derived from the 
French driller, to glitter or sparkle ; and this meaning it 
fully retains when we speak of a brilliant diamond, a 
brilliant star, &c. By what a subtle analogy is it that we 
speak of a brilliant position, a brilliant achievement, 
brilliant talents, brilliant style ! We cannot speak of a 
clear explanation, indefatigable perseverance, perspicuous 
style, or sore calamity, without employing in each of these 
expressions a double analogy to physical impressions, 
actions, or events. It will be shewn in the sixth Lesson 
that to this process we owe the creation of all names 
connected with mental feelings or existences. 

Read Watts' Logic, Chapter iv. 
Locke's Essay on the Hmnan Undei^standingy Book III, 
Chapters IX. and X. 



LESSON V. 

OF THE TWOFOLD MEANING OF TERMS— 
IN EXTENSION AND INTENSION. 

There is no part of the doctrines of Logic to which I 
would more urgently request the attention of the reader 
than to that which I will endeavour to explain clearly in 
the present Lesson. I speak of the double meaning 
which is possessed by most logical terms — the meaning 
in extension, and the meaning in intension. I believe 
that the reader who once acquires a thorough apprehen- 
sion of the difference of these meanings, and learns to 
bear it always in mind, will experience but little further 
difficulty in the study of logic. 

The meaning of a term in extension consists of the 
objects to wMcli the term may "be applied ; its meaning in 
intension consists of the qualities which are necessarily- 
possessed by objects bearing that name. A simple example 
will make this distinction most apparent. What is the 
meaning of the name "metal".'' The first and most ob- 
vious answer is that metal means either gold, or silver, or 
iron, or copper, or aluminium, or some other of the 48 
substances known to chemists, and considered to have a 
metaUic nature. These substances then form the plain 
and common meaning of the name, which is the meaning 
in extension. But if it be asked why the name is applied 
to all these substances and these only, the answer must 
be — Because they possess certain qualities which belong 
to the nature of metal. We cannot, therefore, know to 
what substances we may apply the name, or to what we 



38 TWOFOLD MEANING OF TERMS— 

may not, unless we know the qualities which are indis- 
pensable to the character of a metal. Now chemists lay 
these down to be somewhat as follows: — (i) A metal 
must be an element or simple substance incapable of 
decomposition or separation into simpler substances by 
any known means. (2) It must be a good conductor of 
heat and electricity. (3) It must possess a great and 
peculiar reflective power known as metallic lustre*. 

These properties are common to all metals, or nearly 
all metals, and are what mark out and distinguish a 
metal from other substances. Hence they form in a 
certain way the meaning of the name metal, the meaning 
m intension, as it is called, to distinguish it from the 
former kind of meaning. 

In a similar manner almost any other common name 
has a double meaning. "Steamship" denotes in exten- 
sion the Great Eastern, the Persia, the Himalaya, or any 
one of the thousands of steamships existing or which 
have existed ; in intension it means "a vessel propelled 
by steam-power." Monarch is the name of Queen Vic- 
toria, Victor Emmanuel, Louis Napoleon, or any one of a 
considerable number of persons who rule singly over 
countries ; the persons themselves form the meaning in 
extension ; the quality of ruling alone forms the intensive 
meaning of the name. Animal is the name in extension 
of any one of billions of existing creatures and of indefi- 
nitely greater numbers of other creatures that have ex 
isted or will exist; in intension it implies in all those 
creatures the existence of a certain animal life and sense, 
or at least the power of digesting food and exerting force, 
which are the marks of animal nature. 

* It is doubtfully true that all metals possess metallic lustre, 
and chemists would find it very difticult to give any consistent 
explanation of their use of the name ; but the statements in the 
text are sufficiently true to furnish an example. 



v.] IN EXTENSION AND INTENSION 39 

It is desirable to state here that this distinction of 
extension and intension has been explained by logi- 
cians under various forms of expression. It is the pe- 
culiar misfortune of the science of logic to have a super- 
fluity of names or synonyms for the same idea. Thus the 
intension of a term is synonymous with its comprelien- 
sion, or connotation, or depth; while the extension is 
synonymous with the denotation or breadth. This njay 
be most clearly stated in the form of a scheme : — 

The extension, extent, The intension, intent, 

breadth, denotation, do- depth, connotation, or im- 

main, sphere or application plication of a name con- 

of a name consists of the sists of the qualities the 

individual things to which possession of which by those 

the name applies, things is implied. 

Of these words, denotation and connotation are employed 
chiefly by Mr J. S. Mill among modern logical writers, 
and are very apt fol: the purpose. To denote is to mark 
down^ and the name marks the things to which it may be 
applied or affixed; thus metal denotes gold, silver, cop- 
per, &c. To connote is to mark along with (Latin con^ 
together; notare, to mark), and the connotation accord- 
ingly consists of the qualities before described, the pos- 
session of which is implied by the use of the name metal. 
When we compare different but related terms we may 
observe that they differ in the quantity of their extension 
and intension. Thus the term element has a greater 
extension of meaning than metal, because it includes in 
its meaning all metals and other substances as well. 
But it has at the same time less intension of meaning; 
for among the qualities of a metalhc substance must be 
found the quahties of an element, besides the othei 
qualities peculiar to a metal. If again we compare the 
terms m^tal and malleable metal, it is apparent \ hat the 



40 TWOFOLD MEANING OF TERMS— [less. 

latter term does not include the metals antimony, arsenic, 
and bismuth, which are brittle substances. Hence mal- 
leable metal is a term of narrower meaning in extension 
tha.n metal ; but it has also deeper meaning in intension, 
because it connotes or implies the quality of malleability 
in addition to the general qualities of a metal. White 
7nalleable metal is again a narrower term in extension 
because it does not include gold and copper ; and I can 
go on narrowing the meaning by the use of qualifying ad- 
jectives until only a single metal should be denoted by 
the term. 

The reader will now see clearly that a general law of 
great importance connects the quantity of extension and 
the quantity of intension, viz. — -As the intension of a term 
is increased the extension is decreased. It must not be 
supposed, indeed, that there is any exact proportion be- 
tween the degree in which one meaning is increased and 
the other decreased. Thus if we join the adjective red to 
metal we narrow the meaning much more than if we join 
the adjective white ^ for there are at least twelve times 
as many white metals as red. Again, the term white 
man includes a considerable fraction of the meaning of 
the term man as regards extension, but the term blind 
man only a small fraction of the meaning. Thus it is 
obvious that in increasing the intension of a terra we ma} 
decrease the extension in any degree. 

In understanding this law we must carefully discrimi- 
nate the cases where there is only an apparent increase of 
the intension of a term, from those where the increase is 
real. If I add the term elefnentary to metal, I shall not 
really alter the extension of meaning, for all the metals 
are elements ; and the elementary metals are neither 
more nor less numerous than the metals. But then the 
intension of the term is really unaltered at the same time ; 
for the quality of an element is really found among the 



v.] IN EXTENSION AND INTENSION. 41 

qualities of metal, and it is superfluous to specify it ovei 
again. A quality which belongs invariably to the whole 
of a class of things is commonly called a property of the 
class (see Lesson Xli.), and we cannot qualify or restrict 
a term by its own property. 

This is a convenient place to notice a distinction be- 
tween terms into those which are connotative and those 
which are non-connotative, the latter consisting of the 
terms which simply denote things without implying any 
knowledge of their qualities. As Mr Mill considers this 
distinction to be one of great importance, it will be well 
to quote his own words'^: — 

" A non-connotative term is one which signifies a sub- 
ject only, or an attribute only. A connotative term is 
one which denotes a subject, and implies an attribute. 
By a subject is here meant anything which possesses 
attributes. Thus John, or London, or England, are 
names which signify a subject only. Whiteness, length, 
virtue, signify an attribute only. None of these names, 
therefore, are connotative. But white, ^ojig, virtuous, 
are connotative. The word white denotes all white 
things, as snow, paper, the foam of the sea, &c., and 
implies, or, as it was termed by the schoolmen, cojinotes 
the attribute whiteness. The word white is not predi- 
cated of the attribute, but of the subjects, snow, &c. ; but 
when we predicate it of them, we imply, or connote, that 
the attribute whiteness belongs to them 

"All concrete general names are connotative. The 
word man^ for example, denotes Peter, James, John, and 
an indefinite number of other individuals, of whom, taken 
as a class, it is the name. But it is applied to them, be- 
cause they possess, and to signify that they possess, cer- 



* System of Logic ^ Vol. I. p. 31, 6th ed. Book I. Chap. II, 

§5 



42 TWOFOLD MEANING OF TERMS--- [LESa 

tain attributes. . . . What we call men, are the subjects, 
the individual Styles and Nokes ; not the qualities by 
which their humanity is constituted. The name therefore 
is said to signify the subjects directly, the attributes in- 
directly ; it denotes the subjects, and implies, or involves, 
or indicates, or, as we shall say henceforth, connotes, the 
attributes. It is a connotative name .... 

" Proper names are not connotative : they denote the 
individuals who are called by them ; but they do not indi- 
cate or imply any attributes as belonging to those indivi- 
duals. When we name a child by the name Paul, or a dog 
by the name Caesar, these names are simply marks used 
to cr.aLie those individuals to be made subjects of dis- 
course. It may be said, indeed, that we must have had 
some reason for giving them those names rather than 
any others ; and this is true ; but the name, once given, is 
independent of the reason. A man may have been named 
John, because that was the name of his father; a town 
may have been named Dartmouth, because it is situ- 
ated at the mouth of the Dart. But it is no part of the 
signification of the word John, that the father of the per- 
son so called bore the same name ; nor even of the word 
Dartmouth to be situated at the mouth of the Dart. If 
sand should choke up the mouth of the river, or an earth- 
quake change its course, and remove it to a distance from 
the town, the name of the town would not necessarily be 
changed." 

I quote this in Mr Mill's own words, because though 
it expresses most clearly the view accepted by Mr Mill 
and many others, it is nevertheless probably erroneous. 
The connotation of a name is confused with the etymo- 
logical meaning, or the circumstances which caused it to 
be affixed to a thing^ Surely no one who uses the name 
England, and knows what it denotes, can be ignorant of 
the peculiar qualities and circumstances of the country, 



v.] IN EXTENSION AND INTENSION 43 

and these form the connotation of the term. To any one 
who knows the town Dartmouth the name must imply the 
possession of the circumstances by which that town is cha- 
racterised at the present time. If the river Dart should be 
destroyed or removed, the town would so far be altered, 
and the signification of the name changed. The name 
fvould no longer denote a town situated on the Dart, but 
one which -wdiS formerly situated on the Dart, and it would 
be by a mere historical accident that the form of the name 
did not appear suitable to the town. So again any proper 
name such as John Smith, is almost without meaning until 
we know the John Smith in question. It is true that the 
name alone connotes the fact that he is a Teuton, and 
is a male ; but, so soon as we know the exact individual 
it denotes, the name surely implies, also, the peculiar fea- 
tures, form, and character, of that individual. In fact, as 
it is only by the peculiar qualities, features, or circum- 
stances of a thing, that we can ever recognise it, no name 
could have any fixed meaning unless we attached to it, 
mentally at least, such a definition of the kind of thing 
denoted by it, that we should know whether any given 
thing was denoted by it or not. If the name John Smith 
does not suggest to my mind the qualities of John Smith, 
how shall I know him when I meet him? for he certainly 
does not bear his name written upon his brow *. 

This, however, is quite an undecided question; and 
as Mr Mill is generally considered the best authority upon 
the subject, it may be well for the reader provisionally to 
accept his opinion, that singular or proper names are 
non-connotative, and all concrete general names are con- 
notative. Abstract names, on the other hand, can hardly 



* Further objections to Mr Mill's views on this pouit will 
be found in Mr Shedden*s Elements of Logic, London, 1864. 
pp. 14, &c 



/ 
/ 



44 TWOFOLD MEANING OF TERMS, [less. 

possess connotation at all, for as they already denote the 
attributes or qualities of something, there is nothing left 
which can form the connotation of the name. Mr Mill, 
indeed, thinks that abstract names may often be consi- 
dered connotative, as when the mrnQ fatcli connotes the 
attribute of hurtfulness as belonging to fault. But if 
fault is a true abstract word at all I should regard hurt- 
fulness as a part of its denotation ; I am inclined to think 
\\i7\X faidtiness is the abstract name, and that fault is gene- 
rally used concretely as the name of a particular action or 
thing that is faulty, or possesses faultiness. But the sub- 
ject cannot be properly discussed here, and the reader 
snould note Mr Mill's opinion that abstract names are 
usually non-connotative, but may be connotative in some 
cases. 

The subject of Extension and Intension may be pur- 
sued in Hamilton's Lectures on Logic, Lect. viii. ; 
or in Thomson's Laws of Thought, Sections 48 to 
52. It is much noticed in Spalding's Logic (Ency- 
clopedia Britannica, 8th ed.). 



LESSON VI. 

THE GROWTH OF LANGUAGE. 

Words, we have seen, become equivocal in at least three 
different ways — by the accidental confusion of different 
words, by the change of meaning of a word by its 
habitual association with other things than its original 
meaning, and by analogical transfer to objects of a similar 
nature. We must however consider somewhat more 
closely certain changes in language which arise out of the 



VI.] THE GROWTH OF LANGUAGE, 45 

last cause, and which are in constant progress. We can 
almost trace in fact the way in which language is created 
and extended, and the subject is to the logician one of a 
highly instructive and important character. There are 
two great and contrary processes which modify language 
as follows : — 

1. Generalization, by which a name comes to be 
applied to a wider class of objects than before, so that 
the extension of its meaning is increased, and the inten- 
sion diminished. 

2. Specialization, by which a name comes to be re- 
stricted to a narrower class, the extension being decrease^ 
and the intension increased. 

The first change arises in the most obvious manner, 
from our detecting a resemblance between a new object, 
which is without a name, and some well-known object. 
To express the resemblance we are instinctively led to 
apply the old name to the new object. Thus we are well 
acquainted with glass, and, if we meet any substance 
having the same glassy nature and appearance, we shall be 
apt at once to call it a kind of glass ; should we often meet 
with this new kind of glass it woula probably come to share 
the name equally with the old and original kind of glass. 
The word coal has undergone a change of this kind ; ori- 
ginally it was the name of charked or charred wood, which 
was the principal kind of fuel used five hundred years ago. 
As mineral coal came into use it took the name from the 
fonner fuel, which it resembled more nearly than any- 
thing else, but was at first distinguished as sea-coal or 
pit-coal. Being now far the more common of the two, it 
has taken the simple name, and we distinguish charred 
wood as charcoal. Paper has undergone a like change ; 
originally denoting the papyrus used in the Roman Em- 
pire, it was transferred to the new writing material made 
of cotton or linen rags, which was introduced at a quite 



46 THE GROWTH OF LANGUAGE, [less 

uncertain period. The word character is interesting on 
account of its logical employment; the Greek ;^a/joucr^p 
denoted strictly a tool for engraving, but it became trans- 
ferred by association to the marks or letters engraved 
with it, and this meaning is still retained by the word when 
we speak of Greek characters^ Arabic characters^ i. e. fig^ures 
or letters. But inasmuch as objects often have natural 
marks, signs, or tokens, which may indicate them as well 
as artificial characters, the name was generalized, and now 
means any peculiar or distinctive mark or quality by which 
an object is easily recognised. 

Changes of this kind are usually effected by no parti- 
cular person and with no distinct purpose, but by a sort 
of unconscious instinct in a number of persons using the 
name. In the language of science, however, changes are 
often made purposely, and with a clear apprehension of 
the generalization implied. Thus soap in ordinary life 
is applied only to a compound of soda or potash with 
fat ; but chemists have purposely extended the name 
so as to include any compound of a metallic salt with a 
fatty substance. Accordingly there are such things as 
lime-soap and lead-soap^ which latter is employed in 
making common diachylon plaster. Alcohol at first de- 
noted the product of ordinary fermentation commonly 
called spirits of wine, but chemists having discovered that 
many other substances had a theoretical composition 
closely resembling spirits of wine, the name was adopted 
for the whole class, and a long enumeration of different 
kinds of alcohols will be found in Dr Roscoe's lessons 
on chemistry. The number of known alcohols is likewise 
subject to indefinite increase by the progress of discovery. 
Every one of the chemical terms acid, alkali, metal, alloy, 
earth, ether, oil, gas, salt, may be shown to have under- 
gone great generalizations. 

In other sciences there is hardly a less supply of 



VI.] THE GROWTH OF LANGUAGE, 47 

instances. A lens originally meant a lenticular shaped 
or double convex piece of glass, that being the kind of 
glass most frequently used by opticians. But as glasses 
of other shapes came to be used along with lenses^ the 
name was extended to concave or even to perfectly flat 
pieces of glass. The words lever, plane, cone, cylinder, 
arc, conic section, curve, prism, magnet, pendulum, ray, 
light, and many others, have been similarly generalized. 

In common language we may observe that even 
proper or singular names are often generalized, as when 
in the time of Cicero a good actor was called a Roscius 
after an actor of preeminent talent. The name Caesar 
v/as adopted by the successor of Julius Caesar as an official 
name of the Emperor, with which it gradually became 
synonymous, so that in the present day the Kaisers of 
Austria and the Czars of Russia both take their title from 
Cassar. Even the abstract name Caesarism has been 
formed to express a kind of nnperial system as established 
by Caesar. The celebrated tower built by a king of 
Egypt on the island of Pharos, at the entrance of the 
harbour of Alexandria, has caused lighthouses to be called 
phares in French, and pharos in obsolete EngHsh. From 
the celebrated Roman General Quintus Fabius Maximus 
any one who avoids bringing a contest to a crisis is said 
to pursue a Fabian policy. 

In science also singular names are often extended, as 
when the fixed stars are called distant suns^ or the com- 
panions of Jupiter are called his moons. It is indeed one 
theory, and a probable one, that all general names were 
created by the process of generalization going on in the 
early ages of human progress. As the comprehension of 
general notions requires higher intellect than the appre- 
hension of singular and concrete things, it seems natural 
that names should at first denote individual objects, and 
should afterwards be extended to classes. We have a 



48 THE GROWTH OF LANGUAGE, 

glimpse of this process in the case of the Australian nativor; 
who had been accustomed to call a large dog Cadli, but 
Vv'hen horses were first introduced into the country they 
adopted this name as the nearest description of a horse. 
A very similar incident is related by Captain Cook of the 
natives of Otaheite. It may be objected, however, that a 
certain process of judgment must have been exerted before 
the suitability of a name to a particular thing could have 
been perceived, and it may be considered probable that 
specialization as well as generalization must have acted 
in the earliest origin of language much as it does at 
present. 

Specialization is an exactly opposite process to gene- 
ralization and is almost equally important. It consists in 
narrowing the extension of meaning of a general name, so 
that it comes to be the name only of an individual or a 
mi'nor part of the original class. It is thus we are fur- 
nished with the requisite names for a multitude of new 
implements, occupations and ideas with which we deal in 
advancing civiliza Hon. The name physician is derived 
from the Greek 0uo-t :oj natural, and (pvcnS) nature, so that 
it properly means one who has studied nature, especially 
the nature of the human body. It has become restricted, 
however, to those who use this knowledge for medical 
purposes, and the investigators of natural science have 
been obliged to adopt the new name physicist. The name 
naturalist has been similarly restricted to those who study 
animated nature. The name surgeon originally meant 
handicraftsman, being a corruption of chiritrgeon^ derived 
from the Greek x^^povpycs, hand- worker. It has long been 
specialized however to those who perform the mechanical 
parts of the sanatory art. 

Language abounds with equally good examples. Min- 
ister originally meant a servant, or one who acted as a 
mi7ior of another. Now it often means specially the most 



VI.] THE GROWTH OF LANGUAGE, 49 

important man in the kingdom. A chancellor was a clerk 
or even a door-keeper who sat in a place separated by 
bars or cancelli in the offices of the Roman Emperor's 
palace ; now it is always the name of a hio^h or even the 
highest dignitary. Peer was an equal (Latin, Par), and 
we still speak of being* tried by our peers ; but now, by the 
strange accidents of language, it means the few who are 
superior to the rest of the Queen's subjects in rank. 
Deacon, Bishop, Clerk, Queen, Captain, General, are all 
words which have undergone a like process of specializa- 
tion. In such words as telegraph, rail, signal, station, 
and many words relating to new inventions, we may 
trace the progress of change in a lifetime. 

One effect of this process of specialization is very soon 
to create a difference between any two words which happen 
from some reason to be synonymous. Two or more words 
are said to be synonymous (from the Greek a-vv, with, and 
ovofxa, name) when they have the same meaning, as in the 
case, perhaps, of teacher and instructor, similarity and 
resemblance, beginning and commencement, sameness 
and identity, hypothesis and supposition, intension and 
comprehension. But the fact is that words commonly 
called synonymous are seldom perfectly so, and there are 
almost always shades of difference in meaning or use, 
which are explained in such works as Crabb's English 
Syrionyms, A process called by Coleridge desynonymi- 
zation, and by Herbert Spencer differentiation, is always 
going on, which tends to specialize one of a pair of 
synonymous words to one meaning and the other to 
another. Thus wave and billow originally meant exactly 
the same physical effect, but poets have now appropriated 
the word ^billow,' whereas wave is used chiefly in practical 
and scientific matters. Undulation is a third synonym, 
which will probably become the sole scientific term for 
a wave in course of time. Cab was originally a mere 



so THE GROWTH OF LANGUAGE. [LESS, 

abbreviation of cabriolet, and therefore of similar meaningi 
but it is now specialized to mean almost exclusively a 
hackney cab. In America car is becoming restricted to 
the meaning of a railway car. 

It may be remarked that it is a logical defect in a 
language to possess a great number of synonymous terms, 
since we acquire the habit of using them indifferently 
without being sure that they are not subject to ambiguities 
and obscure differences of meaning. The English lan- 
guage is especially subject to the inconvenience of having 
a complete series of words derived from Greek or Latin 
roots nearly synonymous with other words of Saxon or 
French origin. The same statement may, in fact, be 
put into Saxon or classical English; and we often, as 
Whately has well remarked, seem to prove a state- 
ment by merely reproducing it in altered language. The 
rhetorical power of the language may be increased by the 
''copiousness and variety of diction, but pitfalls are thus 
prepared for all kinds of fallacies. (See Lessons XX 
and XXI .) 

In addition to the effects of generalization and speci- 
alization, vast additions and changes are made in lan- 
guage by the process of analogous or metaphorical exten- 
sion of the meaning of words. This change may be said, 
no doubt, to consist in generahzation, since there must 
always be a resemblance between the new and old appli- 
cations of the term. But the resemblance is often one of 
a most distant and obscure kind, such as we should call 
analogy rather than identity. All words used metapho- 
rically, or as similitudes, are cases of this process of ex- 
tension. The name metaplior is derived from the Greek 
words ii^ra^ over, and cj^epeiv^ to carry ; and expresses ap- 
parently the transference of a word from its ordinary to a 
peculiar purpose. Thus the old similitude of a ruler to 
the pilot of the vessel gives rise to many metaphors, a^ 



VI.] THE GROWTH OF LANGUAGE. 51 

in speaking of the Prime Minister being at the Helm of 
the State. The word governor, and all its derivatives, is, 
in fact, one result of this metaphor, being merely a corrupt 
form oi gubernator^ steersman. The words compass, pole- 
star, ensign, anchor, and many others connected with na- 
vigation, are constantly used in a metaphorical manner. 
From the use of horses and hunting we derive another 
set of metaphors ; as, in taking the reins of government, 
overturning the government, taking the bit between the 
teeth, the Government Whip, being heavily weighted, &c. 
No doubt it might be shewn that every other important 
occupation of life has furnished its corresponding stock 
of metaphors. 

It is easy to shew, however, that this process, besides 
going on consciously at the present day, must have acted 
throughout the history of language, and that we owe to 
it almost all, or probably all, the words expressive of re- 
fined mental or spiritual ideas. The very word spirit^ now 
the most refined and immaterial of ideas, is but the Latin 
spiritus, a gentle breeze or breathing; and inspiration, 
esprit^ or wit, and many other words, are due to this me- 
taphor. It is truly curious, however, that almost all the 
words in different languages denoting mind or soul imply 
the same analogy to breath. Thus, soul is from the 
Gothic root denoting a strong wind or storm ; the Latin 
words anwtus and anhna are supposed to be connected 
with the Greek ave^os, wind ; yjrvxv is certainly derived 
from yjnjx(»), to blow ; TTpevfxa, air or breath, is used in the 
New Testament for Spiritual Being ; and our word ghost 
has been asserted to have a similar origin. 

Almost all the terms employed in mental philosophy 
or metaphysics, to denote actions or phenomena of mind, 
are ultimately derived from metaphors. Apprehension is 
the putting forward of the hand to take anything ; com- 
prehension is the taking of things together in a handful ; 

4—2 



52 THE GROWTH OF LANGUAGE. [LESS 

extension is the spreading out ; intention, the bending to ; 
explication, the unfolding ; application, the folding to ; 
conception, the taking up together ; relation, the carrying 
back ; experience is the thoroughly going through a thing ; 
difference is the carrying apart ; deliberation, the weighing 
out ; interruption, the breaking between ; proposition, the 
placing before ; intuition, the seeing into ; and the list 
might be almost indefinitely extended. Our English 
name for reason, the understanding, obviously contains 
some physical metaphor which has not been fully ex- 
plained ; with the Latin intellect there is also a metaphor. 

Every sense gives rise to words of refined meaning ; 
sapience, taste, insipidity, gout, are derived from the sense 
of taste ; sagacity, from tfie dog's extraordinary power of 
smell ; but as the sense of sight is by far the most acute 
and intellectual, it gives rise to the larger part of lan- 
guage ; clearness, lucidity, obscurity, haziness, perspicuity, 
and innumerable other expressions, are derived from this 
sense. 

It is truly astonishing to notice the power which lan- 
guage possesses by the processes of generalization, speci- 
alization, and metaphor, to create many words from one 
single root. Prof. Max Miiller has given a remarkable 
instance of this in the case of the iz^otspec^ which means 
sights and appears in the Aryan languages, as in the San- 
scrit spas^ the Greek G-KeTrTOfxai, with transposition of con- 
sonants, in the Latin speci'o, and even in the English spy. 
The following is an incomplete list of the words deve- 
loped from this one root ; species, special, especial, speci- 
men, spice, spicy, specious, speciality, specific, specializa- 
tion, specie (gold, or silver), spectre, specification, spec- 
tacle, spectator, spectral, spectrum, speculum, specular, 
speculation. The same root also enters into composi- 
tion with various prefixes; and we thus obtain a series 
of words, suspect, aspect, circumspect, expect, inspect, 



VI.] THE GROWTH OF LANGUAGE, 53 

prospect, respect, retrospect, introspection, conspicuous, 
perspicuity, perspective ; with each of which, again, a 
number of derivatives is connected. Thus, from suspect, 
we derive suspicion, suspicable, suspicious, suspiciously, 
suspiciousness. I have estimated that there are in all 
at least 246 words, employed at some period or other in 
the English language which undoubtedly come from the 
one root spec, 

J. S. Mill's Logic^ Book IV. Chap. V. * On the Natural 
History of the Variations in the Meanings of Terms.* 
Archbishop Trench, On the Study of Words, 
Max Miilier, Lectures on the Science of Language* 



LESSON VII. 

LEIBNITZ ON KNOWLEDGE. 

In treating of terms it is necessary that we should clearly 
understand what a perfect notion of the meaning of a 
term requires. When a name such as monarchy or czvili- 
zatlony or autonomy is used, it refers the mind to some 
thing or some idea, and we ought if possible to obtain 
a perfect knowledge of the thing or idea before we use 
the word. In what does this perfect knowledge consist t 
What "are its necessary characters .f* This is a question 
which the celebrated mathematician and philosopher 
Leibnitz attempted to answer in a small treatise or tract 
first published in the year 1684. This tract has been the 
basis of what is given on the subject in several recent 
works on Logic, and a complete translation of the tract 



54 LEIBNITZ ON KNOWLEDGE, [less. 

has been appended by Mr Baynes to his translation cf 
the Port Royal Logic, As the remarks of Leibnitz him- 
self are not always easy to understand, I will not contine 
myself to his exact words, but will endeavour to give the 
simplest possible statement of his views, according as 
they have been interpreted by Dr Thomson or Sir W, 
Hamilton. 

Knowledge is either obscure or clear ; either confused 
or distinct ; either adequate or inadequate ; and lastly 
either symbolical or intuitive. Perfect knowledge must 
be clear, distinct, adequate and intuitive ; if it fails in any 
one of these respects it is more or less imperfect. We 
may, therefore, classify knowledge as in the following 
scheme : — 





Knowledge. 


Clear. 


1 

Obscure. 


r 

Distinct. 


1 
Confused. 


r 

Adequate. 


1 
Inadequate. 


Intuitive. 


Symbolical. 



Pei-fect. 

A notion, that is to say our knowledge of a thing, is 
obscure when it does not enable us to recognize the thing 
again and discriminate it from all other things. We 
have a clear notion of a rose and of most common flowers 
because we can recognise them with certainty, and do not 
confuse them with each other. Also we have a clear 
notion of any of our intimate friends or persons whom we 
habitually meet, because we recognise them whenever we 
see them with the utmost certainty and without hesita- 
tion. It is said that a shepherd acquires by practice a 
clear notion of each sheep of his flock, so as to enable 
him to single out any one separately, and a keeper of 



VII.] LEIBNITZ ON KNOWLEDGE, 55 

Hounds learns the name and character of each hound, 
while other persons have only an obscure idea of the 
hounds generally, and could not discriminate one from 
the other. But the geologist cannot give a clear idea of 
what sandstone, conglomerate, or schist, or slate, or trap 
rock consists, because different rocks vary infinitely in 
degree and character, and it is often barely possible to 
say w^hether a rock is sandstone or conglomerate, schist 
or slate, and so on. In the lower forms of life the natu- 
ralist hardly has a clear notion o^ animal life, as distin- 
guished from vegetable life ; it is often difficult to decide 
whether a protophyte should be classed with animals or 
plants. 

Clear knowledge, again, is confused, when we cannot 
distinguish the parts and qualities of the thing known, 
and can only recognise it as a whole. Though any one 
instantly knows a friend, and could discriminate him from 
all other persons, yet he would generally find it impos- 
sible to say how he knows him, or by what marks. He 
could not describe his figure or features, but in the very 
roughest manner. A person unpractised in drawing, who 
attempts to delineate even such a familiar object as a 
horse or cow, soon finds that he has but a confused notion 
of its form, while an artist has a distinct idea of the form 
of every limb. The chemist has a distinct as well as a 
clear notion of gold ar d silver, for he can not only tell 
wnth certainty whether any metal is really gold or silver, 
but he can specify and describe exactly the qualities by 
which he knows it ; and could, if necessary, mention a 
great many other qualities as well. We have a very dis- 
tinct notion of a chess-board, because we know it consists 
of 64 square spaces ; and all our ideas of geometrical 
figures, such as triangles, circles, parallelograms, squares, 
pentagons, hexagons, &c. are or ought to be perfectly dis- 
tinct. But when we talk of a constitutional governments 



56 LEIBNITZ ON KNOWLEDGE. [less. 

or a civilized nation, we have only the vaguest idea of 
what we mean. We cannot say exactly what is requisite 
to make a Government constitutional, without including 
also Governments which we do not intend to include ; 
and so of civilized nations; these terms have neither dis- 
tinct nor clear meanings. 

It is to be remarked that no simple idea, such as that 
of red colour^ can be distinct in the meaning here in- 
tended, because nobody can analyse red colour, or de- 
scribe to another person what it is. A person who has 
been blind from birth cannot be made to conceive it ; and 
it is only by bringing an actual red object before the eye 
that we can define its character. The same is generally 
true of all simple sensations, whether tastes, smells, co- 
lours, or sounds; these then may be clearly known, but 
not distinctly^ in the meaning which Leibnitz gives to this 
word. 

To explain the difference which Leibnitz intended to 
denote by the names adequate and inadequate, is not 
easy. He says, "When everything which enters into a 
distinct notion is distinctly known, or when the last ana- 
lysis is reached, the knowledge is adequate, of which I 
scarcely know whether a perfect example can be offered 
— the knowledge of numbers, however, approaches near 
to it." 

To have adequate knowledge of things, then, we must 
not only distinguish the parts which make up our notion 
of a thing, but the parts which make up those parts. For 
instance, we might be said to have an adequate notion of 
a chess-board, because we know it to be made up of 64 
squares, and we know each of those squares distinctly, 
because each is made up of 4 equal right lines, joined 
at right angles. Nevertheless, we cannot be said to have 
a distinct notion of a straight line, because we cannot well 
df'fine it, or resolve it into anything simpler. To be com- 



VII.] LEIBNITZ ON KNOWLEDGE, 57 

pletely adequate, our knowledge ought to admit of analysis 
after analysis ad infinitum^ so that adequate knowledge 
would be impossible. But, as Dr Thomson remarks, we 
may consider any knowledge adequate which carries the 
analysis sufficiently far for the purpose in view. A me- 
chanist, for instance, has adequate knowledge of a ma^ 
chine, if he not only know its several wheels and parts,- 
but the purposes, materials, forms, and actions of those 
parts ; provided again that he knows all the mechanical 
properties of the materials, and the geometrical properties 
of the forms which may influence the working of the 
machine. But he is not expected to go on still further and 
explain why iron or wood of a particular quality is strong 
or brittle, why oil acts as a lubricator, or on what axioms 
the principles of mechanical forces are founded. 

Lastly, we must notice the very important distinction 
of symbolical and intuitive knowledge. From the original 
meaning of the word, intuitive would denote that which 
we gain by seeing (Latin, intueor^ to look at), and any 
knowledge which we have directly through the senses, 
or by immediate communication to the mind, is called 
intuitive. Thus we may learn intuitively what a square 
or a hexagon is, but hardly what a chiliagon, or figure of 
1000 sides, is. 

We could not tell the difference by sight of a figure 
of 1000 sides and a figure of looi sides. Nor can we 
itnagine any such figure completely before the mind. It 
is known to us only by name or symbolically. All large 
numbers, such as those which state the velocity of light 
(186,000 miles per second), the distance of the sun 
(91,000,000 miles), and the like, are known to us only by 
symbols, and they are beyond our powers of imagination. 

Infinity is known in a similar way, so that we can in 
an intellectual manner become acquainted with that oi 
which our senses could never inform us. We speak also 



58 LEIBNITZ ON KNOWLEDGE, [LEsa 

of nothing, of zero, of that which is self-contradictory^ 
of the non-existent, or even of the U7ithinkable or incon- 
ceivable, although the words denote what can never bt* 
reahzed in the mind and still less be perceived through 
the senses intuitively, but can only be treated in a merely 
symbolical way. 

In arithmetic and algebra we are chiefly occupied 
with symbolical knowledge only, since it is not necessary 
in working a long arithmetical question or an algebraical 
problem that we should realise to ourselves at each step 
the meaning of the numbers and symbols. We learn 
from algebra that if we multiply together the sum and 
difference of two quantities we get the difference of the 
squares ; as, in symbols 

{a^b){a''b)^a'^-b^\ 

which is readily seen to be true, as follows • 

a-^b 
a — b 



d^ + ab 
-ab-b^ 

d^ -{-o — b\ 

In the above we act darkly or symbolically, using the 
letters a and b according to certain fixed rules, without 
knowing or caring what they mean ; and whatever mean- 
ing we afterwards give to a and b we may be sure the 
process holds good, and that the conclusion is true with- 
out going over the steps again. 

But in geometry, we argue by intuitive perception of 
the truth of each step, because we actually employ a re- 
presentation in the mind of the figures in question, and 
satisfy ourselves that the requisite properties are really 
possessed by the figures. Thus the algebraical truth 
shown above in symbols may be easily proved to hold true 



vil] LEIBNITZ ON KNOWLEDGE. 59 

of lines and rectangles contained under those lines, as a 
corollary of the 5th Prop, of Euclid's Second Book. 

Much might be said concerning the comparative ad- 
vantages of the intuitive and symbolical methods. The 
latter is usually much the less laborious, and gives the 
most widely applicable answers ; but the symbolical sel- 
dom or never gives the same command and comprehen- 
sion of the subject as the intuitive method. Hence the 
study of geometry is always indispensable in education, 
although the same truths are often more readily proved 
by algebra. It is the peculiar glory of Newton that he 
was able to explain the motions of the heavenly bodies 
by the geometric or intuitive method ; whereas the great- 
est of his successors, such as Lagrange or Laplace, have 
treated these motions by the aid of symbols. 

What is true of mathematical subjects may be applied 
to all kinds of reasoning ; for words are symbols as much 
as A^ B, C, or :r, y, 2, and it is possible to argue with 
words without any consciousness of their meaning. Thus 
if I say that " selenium is a dyad element, and a dyad 
element is one capable of replacing two equivalents of 
hydrogen," no one ignorant of chemistry will be able to 
attach any meaning to these terms, and yet any one will 
be able to conclude that " selenium is capable of replacing 
two equivalents of hydrogen." Such a person argues in a 
purely symbolical manner. Similarly, whenever in com- 
mon life we use words, without having in mind at the 
moment the full and precise meaning of the words, we 
possess symbolical knowledge only. 

There is no worse habit for a student or reader to 
acquire than that of accepting words instead of a know- 
ledge of things. It is perhaps worse than useless to read 
a work on natural history about Infusoria, Foraminifera, 
Rotifera and the like, if these names do not convey clear 
images to the mind. Nor can a student who has not 



6o LEIBNITZ ON KNO W LEDGE. [less. 

vvitnessed experiments, and examined the substances with 
his own eyes, derive any considerable advantage from 
works on chemistry and natural philosophy, where he will 
meet with hundreds of new terms which would be to him 
mere empty and confusing signs. On this account we 
should lose no opportunity of acquamting ourselves, by 
means of our senses, with the forms, properties and 
changes of things, in order that the language we employ 
may, as far as possible, be employed intuitively, and we 
may be saved from the absurdities and fallacies into 
which we might otherwise fall. We should observe, in 
short, the advice of Bacon — ipsis consuescere rebus — 
to accustoin ourselves to things themselves. 

Hamilton's Lectures on Logic. Lect. IX, 
Baynes^ Port Royal Logic, Part i. Chap. 9, and Ao- 
pendix. 



LESSON VIII. 

KINDS OF PROPOSITIONS. 

A TERM standing alone is not capable of expressing truth; 
it merely refers the mind to some object or class of objects, 
about which something may be affirmed or denied, but 
about which the term itself does not affirm or deny any- 
thing. " Sun," " air," " table," suggest to every mind 
objects of thought, but we cannot say that " sun is true,'* 
or " air is mistaken," or " table is false." We must join 
words or terms into sentences or propositions before they 
can express those reasoning actions of the mind to which 



viiL] KINDS OF PROPOSITIONS, 6\ 

tnith or falsity may be attributed. " The sun is bright," 
" the air is fresh," " the table is unsteady," are statements 
which may be true or may be false, but we can certainly 
entertain the question of their truth in any circumstances. 
Now as the logical term was defined to be any combina- 
tion of words expressing an act of simple apprehension, 
so a logical proposition is any combmation of words 
expressing an act of judgment. The proposition is in 
short the result of an act of judgment reduced to the foim 
of language. 

What the logician calls a proposition the grammarian 
calls a sentence. But though every proposition is a sen- 
tence, it is not to be supposed that every sentence is a 
proposition. There are in fact several kinds of sentences 
more or less distinct from a proposition, such as a Sen- 
tence Interrogative or Question, a Sentence Imperative 
or a Command, a Sentence Optative, which expresses a 
wish, and an Exclamatory Sentence, which expresses an 
emotion of wonder or surprise. These kinds of sentence 
may possibly be reduced, by a more or less indirect mode 
of expression, to the form of a Sentence Indicative, which 
is the grammatical name for a proposition; but until this 
be done they have no proper place in Logic, or at least 
no place which logicians have hitherto sufficiently ex- 
plained. 

The name proposition is derived from the Latin wordf 
pro, before, and pono, I place, and means the laying oi 
placing before any person the result of an act of judg- 
ment. Now every act of judgment or comparison must 
involve the two things brought into comparison, and 
every proposition will naturally consist of three parts-^ 
the two terms or names denoting the things compared, 
and the copula or verb indicating the connection between 
them, as it was ascertained in the act of judgment. Thus 
the proposition, " Gold is a yellow substance," expresses 



62 KINDS OF PROPOSITIONS. [less, 

an agreement between gold and certain other substances 
previously called yellow in regard to their colour. Gold 
and yellow substance are evidently the two terms, and ij 
the copula. 

It is always usual to call the first term of a proposi- 
tion the subject, since it denotes the U7tderlying matter, 
as it were (Latin, sub^ under, and jactum, laid) about 
which somethmg is asserted. The second term is called 
the predicate, which simply means that which is affirmed 
or asserted. This name is derived from the Latin prcs- 
(Ticare^ to assert, whence comes the French name p7'edi- 
cateur^ corrupted into our preacher. This Latin verb is 
not to be confused with the somewhat similar one pre- 
die ere y which has the entirely different meaning to pre- 
dict or foretell. I much suspect that newspaper writers 
and others, who pedantically use the verb "to predi% 
cate," sometimes fall into this confusion, and really mean 
to predict^ but it is in any case desirable that a purely 
technical term like predicate should not be needlessly 
introduced into common language, when there are so 
many other good words which might be used. This and 
all other technical scientific terms should be kept to their 
proper scientific use, and the neglect of this rule injures 
at once the language of common life and the language of 
science. 

Propositions are distinguished into two kinds, accord- 
ing as they make a statement conditionally or uncondi- 
tionally. Thus the proposition, *' If metals are heated 
they are softened,^' is conditional, since it does not make 
an assertion concerning metals generally, but only in the 
circumstances when they become heated. Any circum- 
stance which must be granted or supposed before the 
assertion becomes applicable is a condition. Conditional 
propositions are of two kinds. Hypothetical and Disjunc- 
tive., but their consideration will be best deferred to a 



Sentence " 



VIII.] KINDS OF PROPOSITIONS. 63 

subsequent Lesson (xix). Unconditional propositions 
are those with which we shall for some time be solely 
concerned, and these are usually called Categorical Pro- 
positions, from the Greek verb Karrj-yopeco {kategoreo^ to 
assert or affirm). 

The following diagram will conveniently represent the 
classification of sentences and propositions as far as we 
have yet proceeded : — 

Indicative r Categorical . ^ . , 

= Proposition \ ^ .. . , J Hypothetical. 
T .. ^' I Conditional i -r^- • .. ^ 

Interrogative ^ L Disjunctive. 

Imperative 

Optative 

w Exclamatory 

It is nov/ necessary to consider carefully the several 
kinds of categorical propositions. They are classified 
according to quality and according to quantity. As re- 
gards quality they are either afftrmative or negative ; as 
regards quantity they are either universal or particular. 

An affirmative proposition is one which asserts a cer- 
tain agreement between the subject and predicate, so that 
the qualities or attributes of the predicate belong to the 
subject. The proposition, " gold is a yellow substance," 
states such an agreement of gold with other yellow sub- 
stances, that we know it to have the colour yellow, as 
well as whatever qualities are implied in the name sub- 
stance, A negative proposition, on the other hand, as- 
serts a difference or discrepancy, so that some at least of 
the qualities of the predicate do not belong to the sub- 
ject. ^* Gold is not easily fusible" denies that the qua- 
lity of being easily fused belongs to gold. 

Propositions are again divided according to quantity 
into universal and particular propositions. If the propo- 
sition affirms the predicate to belong to the whole of the 
subject, it is an universal proposition, as in the example 



64 KINDS OF PROPOSITIONS. [LEsa 

" all metals are elements," which affirms that the quality 
of being un decomposable or of being simple in nature is 
true of all metals. But if we say " some metals are brit- 
tle," the quality of brittleness is affirmed only of some 
indefinite portion of the metals, and there is nothing in 
the proposition to .make us sure that any certain metal is 
brittle. The name particular being derived from the 
diminutive of the Latin pars would naturally signify a 
small part, but in logic it must be carefully interpreted as 
signifying any part^ from the smallest fraction up to 
nearly the whole. Particular propositions do not include 
cases where a predicate is affirmed of the whole or of 
none of the subject, but they include any between these 
limits. We may accordingly count among particular 
propositions all such as the following: — 

A very few metals are less dense than water. 

Most elements are metals. 

Many of the planets are comparatively small bodies. 

Not a few distinguished men have had distinguished 
sons. 

The reader must carefully notice the somewhat subtle 
point explained further on, that the particular proposition 
though asserting the predicate only of a part of the sub- 
ject, does not deny it to be true of the whole. 

Aristotle, indeed, considered that there were alto- 
gether four kinds of proposition as regards quantity, 

namely — 

[ Universal. 

T» -x- I Particular. 

Proposition < ^. 

Singular. 

^ Indefinite. 

The singular proposition is one which has a singular 
term for its subject, as in — 

Socrates was very wise. 
London is a vast city. 



VIII.] KINDS OF PROPOSITIONS, 65 

But we may fairly consider that a singular proposition 
is an universal one ; for it clearly refers to the whole of 
the subject, which in this case is a single individual thin^. 

Indefinite or indesignate propositions are those which 
are devoid of any mark of quantity whatever, so that the 
form of words gives us no mode of judging whether the 
predicate is applicable to the whole or only part of the 
subject. Metals are useful^ Comets are subject to the law 
of gravitation^ are indefinite propositions. In reality, 
however, such propositions have no distinct place in 
logic at all, and the logician cannot properly treat them 
until the true and precise meaning is made apparent. 
The predicate must be true either of the whole or of part 
of the subject, so that the proposition, as it stands, is 
clearly incomplete ; but if we attempt to remedy this and 
supply the marks of quantity, we overstep the proper 
boundaries of logic and assume ourselves to be acquainted 
with the subject matter or science of which the proposi- 
tion treats. We may safely take the preceding examples 
to mean '^ so7ne metals are useful" and ^' all co7nets are 
subject to the law of gravitation," but not on logical 
grounds. Hence we may strike out of logic altogether 
the class of indefinite propositions, on the understanding 
that they must be rendered definite before we treat them. 
I may observe, however, that in the following lessons I 
shall frequently use propositions in the indefinite form as 
examples, on the understanding that where no sign of 
quantity appears, the universal quantity is to be assumed. 
It is probable that wherever a term is used alone, it 
ought to be interpreted as meaning the whole of its class. 
But however this may be, we need not recognize the inde- 
finite proposition as a distinct kind ; and singular propo- 
sitions having been resolved into universals, there remain 
snly the two kinds, Universal and Particular. 

Remembering now that there are two kinds of propo* 



66 KINDS OF PROPOSITIONS. [less. 

sition as regards quality, and two as regards quantity^ we 
shall be able to form altogether four varieties, thus : — 



Proposition - 



Universal (^,«^™=*''^® ^ 

L Negative E 

Particular (^,«^''™^''^«^ \ 

L Negative 



The vowel letters placed at the right hand are sym- 
bols or abbreviated names, which are always used to 
denote the four kinds of proposition; and there will be 
no difficulty in remembering their meaning if we observe 
that A and I occur in the Latin verb affirmo, I affirm, and 
E and in nego, I deny. 

There will not generally be any difficulty in referring 
to its proper class any proposition that we meet with in 
writings. The mark of universality usually consists of 
some adjective of quantity, such as all, every, each, any, 
the whole J but whenever the predicate is clearly intended 
to apply to the whole of the subject we may treat the pro- 
position as universal. The signs of a particular proposi- 
tion are the adjectives of quantity, some, certain, a few, 
many, most, or such others as clearly indicate part at 
least. 

The negative proposition is known by the adverbial 
particle not being joined to the copula; but in the propo- 
sition E, that is the universal negative, we frequently use 
the particle no or none prefixed to the subject. Thus, 
" no metals are compound," " none of the ancients were 
acquainted with the laws of motion," are familiar forms of 
the universal negative. 

The student must always be prepared too to meet with 
misleading or ambiguous forms of expression. Thus the 
proposition, " all the metals are not denser than water," 
might be taken as E or 0, according as we interpret it to 



VIII.] KINDS OF PROPOSITIONS. 6? 

mean " no metals are denser than water,'' or " not all 
the metals," &c., the last of course being the true sense. 
The little adjective few is very subject to a subtle am- 
biguity of this kind ; for \i I say ^'few books are at once 
learned and amusing," I may fairly be taken to assert 
that a few books certainly are so, but what I really mean 
to draw attention to is my belief that 'Hhe greater num- 
b*:r of books are not at once learned and amusing." A 
proposition of this kind is generally to be classed rather 
as than I. The word some is subject to an exactly 
similar ambiguity between some but not all, and some at 
least, it may be all; the latter appears to be the correct 
interpretation, as shewn in the following lesson (p 79). 

As propositions are met with in ordinary language 
they are subject to various inversions and changes of the 
simple logical form. 

(i) It is not uncommon, especially in poetry, to find 
the predicate placed first, for the sake of emphasis or 
variety ; as in " Blessed are the merciful ;" " Comes some- 
thing down with eventide ;" " Great is Diana of the Ephe- 
sians." There is usually no difficulty in detecting such 
an inversion of the terms, and the sentence must then 
be reduced to the regular order before being treated in 
logic. 

(2) The subject may sometimes be mistaken for the 
predicate when it is described by a relative clause, stand- 
ing at the end of the sentence, as in "no one is free who 
is enslaved by his appetites." Here free is evidently 
the predicate, although it stands in the middle of the 
sentence, and "one who is enslaved by his appetites'' 
is the real subject. This proposition is evidently of the 
form E. 

Propositions are also expressed in various modes dif- 
fering from the simple logical order, and some of tiie 
different kinds which arise must be noticed. 

5—2 



63 KINDS OF PROPOSITIONS. [LESS. 

Exclusive propositions contain some words, such as 
only^ alone, none but, which limit the predicate to the 
subject. Thus, in "elements alone are metals," we are 
informed that the predicate "metal" cannot be applied to 
anything except "elements," but we are not to understand 
that " all elements are metals." The same meaning is 
expressed by " none but elements are metals ;'' or, again, 
by " all that are not elements are not metals ;" and this we 
shall see in the next lesson is really equivalent to "all 
metals are elements." Arguments which appear fallacious 
at first sight will often be found correct when they con- 
tain exclusive propositions and these are properly inter- 
preted. 

Exceptive propositions affirm a predicate of all the 
subject with the exception of certain defined cases, to 
which, as is implied, the predicate does not belong. Thus, 
" all the planets, except Venus and Mercury, are beyond 
the earth's orbit," is a proposition evidently equivalent to 
two, viz. that Venus and Mercury are not beyond the 
earth's orbit, but that the rest are. If the exceptions 
are not actually specified by name an exceptive proposi- 
tion must often be treated as a particular one. For if 
I say " all the planets in our system except one agree with 
Bode's law," and do not give the name of that one excep- 
tion, the reader cannot, on the ground of the proposition, 
assert of any planet positively that it does agree with 
Bode's law. 

Some propositions are distinguished as explicative or 
essential, because they merely affirm of their subject a 
predicate which is known to belong to it by all who can 
define the subject. Such propositions merely unfold 
what is already contained in the subject. "A parallelo- 
gram has four sides and four angles," is an explicative or 
essential proposition. "London, which is the capital of 
England, is the largest city of Europe," contains two pra- 



VIII.] KINDS OF PROPOSITIONS. 69 

positions ; of which one merely directs our attention to 
a fact which all may be supposed to know, viz. that 
London is the capital of England. 

Ampliative propositions, on the other hand, join a 
new predicate to the subject Thus to those who do not 
know the comparative sizes of cities in Europe, the last 
example contains an ampliative proposition. The greater 
number of propositions are of this kind. 

Tautologous or Truistic propositions are those which 
merely affirm the subject of itself, and give no informa- 
tion whatever; as in, "whatever is, is;" "what I have 
written, I have written." 

It is no part of formal Logic to teach us how to inter- 
pret the meanings of sentences as we meet them in writ- 
ings; this is rather the work of the grammarian and 
philologist. Logic treats of the relations of the different 
propositions, and the inferences which can be drawn from 
them; but it is nevertheless desirable that the reader 
should acquire some familiarity with the real logical 
meaning of conventional or peculiar forms of expression, 
and a number of examples will be found at the end of 
the book, which the reader is requested to classify and 
treat as directed. 

In addition to the distinctions already noticed it has 
long been usual to distinguish propositions as they are 
pure or modal. The pure proposition simply asserts that 
the predicate does or does not belong to the subject, while 
the modal proposition states this cum modo, or with an 
intimation of the mode or manner in which the predicate 
belongs to the subject. The presence of any adverb of 
time, place, manner, degree, &c., or any expression equi- 
valent to an adverb, confers modality on a proposition. 
"Error is always in haste;" "justice is ever equal;" "a 
perfect man ought always to be conquering himself," are 
examples of modal propositions in this acceptation of 



70 KINDS OF PROPOSITIONS. [less. 

the name. Other logicians, however, have adopted a 
different view, and treat modality as consisting in the 
degree of certainty or probability with which a judgment 
is made and asserted. Thus, we may say, " an equilateral 
triangle is necessarily equiangular ;" " men are generally 
trustworthy f ** a falling barometer probably indicates a 
coming storm;" "Aristotle's lost treatises may possibly be 
recovered ;" and all these assertions are made with a dif- 
ferent degree of certainty or modality. Dr Thomson is 
no doubt right in holding that the modality does not 
affect the copula of the proposition, and the subject could 
only be properly treated in a work ori Probable Reason- 
ing. 

Many logicians have also divided propositions ac- 
cording as they are true or false, and it might well seem 
to be a distinction of importance. Nevertheless, it is 
wholly beyond the province of the logician to consider 
whether a proposition is true or not in itself; all that he 
has to determine is the comparative truth of propositions 
— that is, whether one proposition is true when another 
is. Strictly speaking, logic has nothing to do with a pro- 
position by itself; it is only in converting or transmuting 
certain propositions into certain others that the work of 
reasoning consists, and the truth of the conclusion is only 
so far in question as it follows from the truth of what we 
shall call the premises. It is the duty of the special sci- 
ences each in its own sphere to determine what are true 
propositions and what are false, and logic would be but 
another name for the whole of knowledge could it take 
this duty on itself. 

See Mr Mill's System of Logtc^ Book I. Chap, iv, 
which generally agrees with what is given above. Chap- 
ters V. and VI. contain Mr Mill's views on the Nature 
and Import of Propositions, which subject may be further 



IX.] THE OPPOSITION OF PROPOSITIONS, 71 

studied in Mr Mill's Examination of Sir W, Hamiltott^s 
Philosophy^ Chap. xvni. ; Hamilton's Lectures on Logic^ 
No. XIII. ; and MansePs Prolegomena Logica^ Chap. il. ; 
but the subject is too metaphysical in character to be 
treated in this work. 



LESSON IX. 

THE OPPOSITION OF PROPOSITIONS. 

We have ascertained that four distinct kinds of propo- 
sitions are recognized by logicians, — the Universal affirm- 
ative, the Particular affirmative, the Universal negative, 
and the Particular negative, commonly indicated by the 
symbols A, I, E, 0. It is now desirable to compare toge- 
ther somewhat minutely the meaning and use of proposi- 
tions of these various kinds, so that we may clearly learxi 
how the truth of one will affect the truth of others, or how 
the same truth may be thrown into various forms of ex- 
pression. 

The proposition A expresses the fact that the thing or 
class of things denoted by the subject is included in, and 
forms part of the class of things denoted by the predicate. 
Thus "all metals are elements" means that metals form 
a part of the class of elements, but not the whole. As 
there are altogether 63 known elements, of which 48 are 
metals, we cannot say that all elements are metals. The 
proposition itself does not tell us anything about ele7nents 
in general; it is not in fact concerned with elements, 
metals being the subject about which it gives us informa- 



mmnk 



72 THE OPPOSITION [less. 

tion. This is best indicated by a kind of diagramj first 
used by the celebrated mathematician Euler, in his letters 
to a German princess. In Fig. i, the metals are supposed 
to be enclosed in the small circle somewhat as sheep 
might be in a pinfold, this circle containing all the metals 
and nothing else. The greater circle is supposed to con- 
tain in a similar manner all the elements and nothing 
else. Now as the small circle is wholly within the larger 
one, it follows that all the metals must be counted as 

Fig. I. 




elements, but of the part of the elements outside the 
circle of metals we know nothing from the proposition. 

The particular afiarmative proposition I exactly resem- 
bles A in meaning, except that only part of the subject is 
brought into question. When I say that " some metals 
are brittle," I mean that of a collection of all the dif- 
ferent metals a few at least might be picked out which 
would be found to be brittle ; but the word some is ex- 
ceedingly indefinite, shewing neither the exact number of 
brittle metals, nor how we are to know them from the 
others, unless indeed by trying whether they are brittle. 
This proposition will be properly represented in Euler's 
mode by two intersecting circles, one supposed to enclose 
all metals, and the other all brittle substances. The 
mere fact of the two circles intersecting proves that some 



IX.] 



OF PROPOSITIONS. 
Fig. a. 



73 




part of one class must coincide with some part of the 
other class, which is what the proposition is intended to 
express. Concerning the portions of the circles which do 
not overlap the proposition tells us nothing. 

The universal negative proposition E denies the ex- 
istence of any agreement or coincidence between the sub- 
ject and predicate. Thus from " no metals are compound 
substances/' we learn that no metal is to be found among 
compound substances, and it follows necessarily that no 
compound substance can be found among the metals. 
For were there a compound substance among the metais, 
there would evidently be one metal at least among the 
compound substances. This entire separation in thought 
of the t^^o classes is well shewn in Euler's method by 
two disconnected circles. 

Fig. 3. 




The reader will easily see that the proposition E is 



74 '^^^ OPPOSITION [LESS. 

distinguished from A and I, by the fact that it gives us 
some information concerning the whole of the predicate^ 
because we learn that none of the objects included in the 
predicate can be found among those included in the sub- 
ject. The affirmative propositions, on the other hand, 
warranted us in holding that the objects denoted by the 
subject, or some particular part of them, were included in 
the predicate, but they give us no warrant for saying 
that any specified part of the predicate is in the subject 
Because we merely know that a substance is an element, 
we do not learn from the proposition " all metals are ele- 
ments" whether it is a metal or not. And from the pro- 
position " some metals are brittle,^' we certainly cannot 
ascertain whether any particular brittle substance is a 
metal. We must seek the information from other sources. 
But from *'no metals are compounds" we learn of any 
compound substance that it is not a metal, as well as of 
a metal that it is not a compound substance. 

The important difference above explained is expressed 
in technical language by saying that the proposition E 
distributes its predicate^ whereas the affirmative proposi- 
tions A and I do not distribute their predicates. By dis- 
tribution of a term is simply meant taking it universallyy 
or referring to all parts of it; and as the validity of any 
argument or syllogism will usually depend upon the suffi- 
cient distribution of the terms occurring in it, too much 
attention cannot be paid to this point. 

Judging from the examples we have had, it will be 
seen that the universal affirmative distributes its subject, 
but not its predicate ; for it gives us some information 
concerning all metals, but not all elements. The parti- 
cular affirmative distributes neither subject nor predicate; 
for we do not learn anything from our example concern- 
ing all metals nor concerning all brittle substances. But 
the universal negative distributes both subject and predi- 



vL] 



OF PROPOSITIONS. 



75 



catc, for we learn something of all tmials and also of all 
compound substances. 

The particular negative proposition will be found to 
distribute its predicate, but not its subject. When I say 
some metals are not brittle^ I intentionally refer only to 
a part of the metals, and exclude them from the class 
Qi brittle substances ; but I cannot help at the same time 
referring to the whole of the brittle substances. If the 
metals in question coincided with any part of the brittle 
substances they could not be said to be excluded from 
the class. To exclude a thing from any space, as from 
a particular chamber of a house, it must not merely be 
removed from some part, but from any part, or from the 
whole of that space or chamber. Euler's diagram for 
this proposition may be constructed in the same manner 
as for the proposition I as follows : — 

Fig. 4. 




It is apparent that though part of the metals fall into 
the circle of brittle substances, yet the remaining portion 
are excluded from any part of the predicate. 

We may state the result at which we have now arrived 
in the following form : — 



o 









Universal \ Affinnativc A. 
( Negative E. 

Tt _j.- ^ \ Affirmative I. 
Particular < ^. . 

{ Negaave 0. 



Subject. 

Distributed. 

Distributed. 

Undistributed. 
Undistributed, 



Predicate. 

Undistributed. 

Distributed. 

Undistributed. 
Distributed- 



76 THE OPPOSITION [less. 

We shall now discover with great case the relations of 
the four propositions, each to each, that is to say, the v/ay 
in which they are opposed to each other. It is obvious 
that the truth of one proposition interferes more or less 
completely with the truth of another proposition having 
the same subject and predicate. If " all metals are ele- 
ments," it is impossible that ^^ some metals are not ele- 
ments,'' and still more palpably impossible, so to say, that 
" no metals should be elements." The proposition A, then, 
is inconsistent with both E and ; and, vice versd, E and 
are inconsistent with A. Similarly, E is inconsistent 
with A and I. But this important difference must be noted, 
that if A be false, is necessarily true, but E may or may 
not be true. If it is not true that "all men are sincere," 
it follows that " some men are not sincere," but it does 
not in the least follow that " no men are sincere." This 
difference is expressed by saying that A and are con- 
tradictory propositions, whereas A and E are called con- 
trary propositions. It is plain that A and E, as in " all 
men are sincere " and " no men are sincere," represent 
the utmost possible contrariety of circumstances. In 
order to prove the falsity of A, it is sufficient to establish 
the truth of 0, and it is superfluous, even if possible, to 
prove E ; similarly E is disproved by proving I, and it 
is superfluous to prove A. Any person who asserts a uni- 
versal proposition, either A or E, lays himself under the 
necessity of explaining away or disproving every single 
exception brought against it. An opponent may always 
restrict himself to the much easier task of finding in- 
stances which apparently or truly contradict the univer- 
sality of the statement, but if he takes upon himself to 
affirm the direct contrary, he is himself open to easy at- 
tack. Were it to be asserted, for instance, that "All 
Christians are more moral than Pagans," it would be 
easy to adduce examples showing that " Some Christians 



IX] OF PROPOSITIONS. 77 

are not more moral than Pagans," but it would be absurd 
to suppose that it would be necessary to go to the con- 
trary extreme, and shew that "No Christians are more 
moral than Pagans." In short A is sufficiently and best 
disproved by 0, and E by I. It will be easily apparent 
that, vice versa, is disproved by A, and I by E ; nor is 
there, indeed, any other mode at all of disproving these 
particular propositions. 

When we compare together the propositions I and 
we find that they are in a certain sense contrary in na- 
ture, one being affirmative and the other negative, but 
that they are still consistent with each other. It is true 
both that " Some metals are brittle," for instance Anti- 
mony, Bismuth and Arsenic ; but it is also true that 
" Some metals are not brittle." And the reader will ob« 
serve that when I affirm " Some metals are elements," 
there is nothing in this to prevent the truth of " Some 
metals are not elements," although on other grounds we 
know that this is not true. The propositions I and are 
called subcontraries each of the other, the name con- 
noting a less degree of contrariety than exists between A 
and E. 

As regards the relation of A to I and E to 0, it is plain 
that the truth of the unive»*sal includes and necessitates 
the truth of the particular What may be affirmed or 
denied of all parts of a class may certainly be affirmed or 
denied similarly of some part of the class. From the 
truth of the particular we have no right to infer either 
the truth or falsity of the universal of the same quality. 
These pairs of propositions are called subalterns, i. e. 
one under the other (Latin sub under, and alter the other 
of two), or we may say more exactly that I and are 
respectively the subalternates of A and E, each of which 
is a subalternans. 



78 



THE OPPOSITION 



[LESS. 



The relations of the propositions just described arc 
all clearly shown in the following scheme : — 



A Contraries 



CO 



& 






i^-' 



w 

>^ 

•4-* 

f— « 

CO 



Subcontraries 



It is so highly important to appnehend completely and 
readily the consistency or opposition of propositions, that 
I will put the matter in another form. Taking any two 
propositions having the same subject and predicate, they 
must come under one of the following statements : — 

1. Of contradictory propositions, one must be true 
and one false. 

2. Of contrary propositions, both cannot be true, and 
both may be false. 

3. Of subcontrary propositions, one only can be false, 
and both may be true. 

4. Of subalterns, the particular is true if the universal 
be true ; but the universal may or may not be true when 
the particular is true. 

I put the same matter in yet another form in the fol- 
lowing table, which shows how the truth of each of A, ^ 
I, and 0, affects the truth of each of the ethers. 



IX.] OF PROPOSITIONS. 79 





A 


E 


I 







is 


is 


is 


is 


If A be true 


true 


false 


true 


false. 


)» ^ 9J » 


false 


true 


false 


true. 


» ^ « » 


doubtful 


false 


true 


doubtful. 


j> ^ » » 


false 


doubtful 


doubtful 


true. 



It will be evident that from the affirmation of univer- 
sal more information is derived than from the affirmation 
of particulars. It follows that more information can be 
derived from the denial of particulars than from the 
denial of universals, that is to say, there are less cases left 
doubtful, as in the above table. 

The reader may well be cautioned, however, against 
an ambiguity which has misled some even of the most 
eminent logicians. In particular propositions the adjcc 
tive some is to be carefully interpreted as sojne, and there 
may or may not be more or alL Were we to interpret it 
as seme, not more nor all, then it would really give to the 
proposition the force of I and combined. If I say " some 
men are sincere," I must not be taken as implying that 
"some men are not sincere;" I must be understood to 
predicate sincerity of some men, leaving the character of 
the remainder wholly unaffected. It follows from this 
that, when I deny the truth of a particular, I must not be 
understood as implying the truth of the universal of the 
same quality. To deny the truth of " some men are mor- 
tal'* might seem very natural, on the ground that not some 
but all men are mortal ; but then the proposition denied 
would really be so7ne tnen are not mortal, i. e. not I. 
Hence when I deny that "some men are immortal" I 
mean that "no men are immortal ;" and when I deny that 
" some men arc not mortal,'^ I mean that " all men are 
mortal." 

It has long been usual to compare propositions aa 



8o OPPOSITION OF PROPOSITIONS, [less. ix. 

regards the quality of the subject matter to which they 
refer, and what is technically called the matter was dis- 
tinguished into three kinds, necessary, contingent, and Im- 
possible. Necessary matter consists of any subject in 
which the proposition A may be affirmed ; impossible in 
which E may be affirmed. Any subject or branch of know- 
ledge in which universal statements cannot usually be 
made is called contingent matter, and it implies the truth 
of I and O. Thus "comets are subject to gravitation," 
though an indefinite or indesignate proposition (p. 65), 
may be interpreted as A, because it refers to a part of 
natural science where such general laws obtain. But 
"men are sincere" would be properly interpreted as par- 
ticular or I, because the matter is clearly contingent. The 
truth of the following statements is evident. 

In necessary matter A and I are true ; E and false. 
In contingent matter I and are true ; A and E false. 
In impossible matter E and are true ; A and I false. 

In reality, however, this part of logical doctrine is 
thoroughly illogical, because in treating a proposition we 
have no right, as already explained (p. 70), to assume 
ourselves acquainted with the science to which it refers. 
Our duty is to elicit the exact consequences of any state- 
ments given to us. We must learn in logic to transform 
information in every possible way, but not to add extra- 
Dcous facts. 



LESSON X. 

CONVERSION OF PROPOSITIONS, AND 
IMMEDIATE INFERENCE. 

We are said to infer whenever we draw one truth 
from another truth, or pass from one proposition to 
another. As Sir W. Hamilton says, Inference is " the 
carrying out into the last proposition what was virtually 
contained in the antecedent judgments.*' The true 
sphere of the science of logic indeed is to teach the 
principles on which this act of inference must be per- 
formed, and all the previous consideration of terms 
and propositions is only useful or pertinent so far as 
it assists us to understand the processes of inference. 
We have to consider in succession all the modes in 
which the same information may be moulded into differ- 
ent forms of expression often implying results of an 
apparently different character. Logicians are not agreed 
exactly as to what we may include under the name 
Inference, and what we should not. All would allow 
that there is an act of inference when we see drops ot 
water on the ground and believe that it has rained. 
This is a somewhat complicated act of inference, which 
we shall consider in later lessons under the subject of 
Induction. Few or none would say that there is an act 
of inference in passing from "The Duke of Cambridge 
is the Commander-in-chief," to "The Commander-in- 
chief is the Duke of Cambridge." But without paying 
much regard to the name of the process I shall in this 



82 CONVERSION OF PROPOSITIONS, [lksS. 

lesson point out all the ways in which we can from a 
single proposition of the forms A, E, I or 0, pass to another 
proposition. 

We are said to convert a proposition when we 
transpose its subject and predicate ; but in order that 
the converse or converted proposition shall be inferred 
from the convertend, or that which was to be converted, 
we must observe two rules (i) the quality of the pro- 
position (affirmative or negative) must be preserved, and 
(2) no term must be distributed in the Converse unless it 
was distributed in the Convertend, 

If in "all metals are elements" we were simply to 
transpose the terms, thus — " all elements are metals," we 
imply a certain knowledge about all elements, whereas 
it has been clearly shewn that the predicate of A is un- 
distributed, and that the convertend does not really give 
us any information concerning all elements. All that 
we can infer is that "some elements are metals;" this 
converse proposition agrees with the rule, and the pro- 
cess by which we thus pass from A to I is called Con- 
version by Limitation, or Per accidens. 

When the converse is a proposition of exactly the 
same form as the convertend the process is called simple 
conversion. Thus from " some metals are brittle sub- 
stances" I can infer "some brittle substances are 
metals," as all the terms are here undistributed. Thus 
I is simply converted into I. 

Again, from " no metals are compounds,** I can pass 
directly to "no compounds are metals," because these 
propositions are both in E, and all the terms are there- 
fore distributed. Euler's diagram (p. y"^^ Fig. 3) clearly 
shows, that if all the metals are separated from all the 
compounds, all the compounds are necessarily separatee 
from all the metals. The proposition E is then simply 
converted into E. 



X.] AND IM MEDIA TE INFERENCE. 85 

But in attempting to convert the proposition we 
encounter a peculiar difficulty, because its subject is un- 
distributed; and yet the subject should become by con- 
version the predicate of a negative proposition, which 
distributes its predicate. Take for example the propo- 
sition, "some existing things are not material substances." 
By direct conversion this would become "all material 
substances are not existing things ;" which is evidently 
absurd. The fallacy arises from existing things being 
distributed in the converse, whereas it is particular in 
the convertend ; and the rules of the Aristotelian logic 
prevent us from inserting the sign of particular quantity 
before the predicate. The converse would be equally 
untrue and fallacious were we to make the subject par- 
ticular, as in " some material substances are not exist- 
ing things." We must conclude, then, that the propo- 
sition cannot be treated either by simple conversion or 
conversion by limitation. It is requisite to apply a new 
process, which may be called Conversion by Negation, 
and which consists in first changing the convertend into 
an affirmative proposition, and then converting it simply. 
If we attach the negation to the predicate instead of 
to the copula, the proposition becomes "some exist- 
ing things are immaterial substances," and, converting 
simply, we have — "some immaterial substances are ex- 
isting things," which may truly be inferred from the con- 
vertend. The proposition 0, then, is only to be converted 
by this exceptional method of negation. 

Another process of conversion can be applied to the 
proposition A, and is known as conversion by contra- 
position. From "all metals are elements," it neces- 
sarily follows that "all not-elements are not metals.^- 
If this be not at the first moment apparent, a little re- 
flection will render it so, and from fig. 5 we see that if 
all the metals be among the elements, whatever is not ele- 

6—2 



84 CONVERSION OF PROPOSITIONS, [less 

Kient, or outside the circle of elements, must also be 
outside the circle of metals. We may also prove the truth 

Fig. 5. 




of the contrapositive proposition in this way, if we may 
anticipate the contents of Lesson xxili.: — If what is not- 
element should be metal, then it must be an element by 
the original proposition, or it must be at once an ele- 
ment and not an element ; which is impossible accord- 
ing to the Primary Laws of Thought (Lesson Xiv.), since 
nothing can both have and not have the same property. 
It follows that what is not-element must be not-metal. 

Mistakes may readily be committed in contrapositive 
conversion, from a cause which will be more apparent in 
Lesson xxii. We are very liable to infer from a pro- 
position of the form "all metals are elements," that all 
not-metals are not-elenients^ which is not only a false 
statement in itself, but is not in the least warranted by 
the original proposition. In fig. 5, it is apparent that 
because a thing lies outside the circle of metals, it does 
not necessarily lie outside the circle of elements, which is 
wider than that of metals. Nevertheless the mistake is 
often made in common life, and the reader will do well 
to remember that the process of conversion by contra- 
position consists only in taking the negative of the pre< 
dicate of the proposition A, as a new subject, and affirm- 
ing of it universally the negative of the old subject 



X.] AND IMMEDIA TE INFERENCE, 85 

Contrapositive conversion cannot be applied to the 
particular propositions I and at all, nor to the propo- 
sition E, in that form ; but we may change E into A by 
attaching the negation to the predicate, and then the 
process can be applied. Thus "no men are perfect," 
may be changed into "all men are not-perfect," i.e. 
"are imperfect," and then we infer by contraposition 
" all not-imperfect beings are not-men." But not-im- 
pe7'fect is really the same as perfect, so that our new 
proposition is really equivalent to " all perfect beings are 
not men,'* or " no perfect beings are men," (E) the sim- 
ple converse of the original proposition. 

There remain to be described certain deductions 
which may be drawn from a proposition without convert- 
ing its terms. They may be called immediate inferences, 
and have been very clearly described by Archbishop 
Thomson in his " Outline of the Necessary Laws of 
Thought "(pp. 156, &c.). 

Immediate Inference t>y Privative Conception consists 
in passing from any affirmative proposition to a negative 
proposition implied in it, or equivalent to it, or vice versa, 
in passing from a negative proposition to its correspond- 
ing affirmative. 

The following table contains a proposition of each 
kind changed by privative conception into an equivalent 
proposition : 

{A all metals are elements. 
E no metals are compounds. 
JE no men are perfect. 
(A all men are imperfect. 
I some men are trustworthy. 
some men are not untrustworthy. 

some men are not trustworthy. 

1 some men are untrustworthy. 
The truth of any of the above can be clearly illustrated 






56 CONVERSION OF PROPOSITIONS, [less, 

by diagrams ; thus it will be apparent that if the whole 
circle of metals lies inside the circle of elements, no part 
can lie outside of that circle or among the compounds. 
Any of the above propositions may be converted, but the 
results will generally be such as we have already ob- 
tained. Thus the simple converse of " no metals are 
compounds" is "no compounds are metals," or "no not- 
elements are metals," the contrapositive of "all metals 
are elements." From the last example we get also by 
simple conversion " some untrustworthy beings are men,'' 
which is obviously the converse by negation, as before 
explained. Applying this kind of conversion to " some 
men are not untrustworthy," we have " some not-untrust- 
worthy beings are men." Lastly, from "all men are 
imperfect" we may obtain through conversion by limita- 
tion, " some imperfect beings are men." 

Immediate Inference toy added determinants consists 
in joining some adjective or similar qualification both to 
tne subject and predicate of a proposition, so as to ren- 
der the meaning of each term narrower or better deter- 
mined. Provided that no other alteration is made the 
truth of the new proposition necessarily follows from the 
truth of the original in almost all cases. 

From "all metals are elements," we may thus infer 
that " all very heavy metals are very heavy elements.'' 
From "a comet is a material body" we infer "a visible 
comet is a visible material body." But if we apply this 
kind of inference too boldly we may meet with fallacious 
and absurd results. Thus, from "all kings are men," 
we m^ight infer " all incompetent kings are incompetent 
men ;" but it does not at all follow that those who are 
incompetent as kings would be incompetent in other 
positions. In this case and many others the qualifying 
adjective is liable to bear different meanings in the sub- 
ject and predicate; but the inference will only be true ol 



X.] AND IMMEDIATE INFERENGE, 87 

necessity when the meaning is exactly the same in each 
case. With comparative terms this kind of inference 
will seldom be applicable; thus from "a cottage is a 
building," we cannot infer "a huge cottage is a huge 
building/' since a cottage may be large when compared 
with other cottages, but not with buildings generally. 

Immediate Inference by Complex Conception is closely 
similar to the last, and consists in employing the subject 
and predicate of a proposition as parts of a more com- 
plex conception. From " all metals are elements," I can 
pass to " a mixture of metals is a mixture of elements." 
From "a horse is a quadruped" I infer *'the skeleton of 
a horse is the skeleton of a quadruped." But here again 
the reader must beware of applying the process where 
the new complex conception has a different meaning in 
the subject and predicate. Thus, from " all Protestants 
are Christians,'^ it does not follow that " a majority of 
Protestants are a majority of Christians," nor that "the 
most excellent of the Protestants is the most excellent of 
the Christians.'^ 

The student is recommended to render himself fami- 
liar with all the transformations of propositions, or im- 
mediate inferences described in this lesson ; and copious 
examples are furnished for the purpose. It is a good 
exercise to throw the same proposition through a series 
of changes, so that it comes out in its original form at 
last, and thus proves the truth of all the intermediate 
changes ; but should conversion by limitation have been 
used, the original universal proposition cannot be re- 
gained, but only the particular proposition corresponding 
to it. 

On I?n7nediate Inference^ Archbishop Thomson, 
Outline of the Laws of Thought^ §§ 85 — 92. 



LESSON XL 

LOGICAL ANALYSIS OF SENTENCES. 

Propositions as they are usually to be found in writ* 
ten or spoken compositions seldom exhibit the simple 
form, the conjunction of a subject, copula, and predicate, 
which we have seen to be the proper logical construction. 
Not only is the copula often confused with the predicate, 
but several propositions may be combined into one gram- 
matical sentence. For a full account of the analysis 
of sentences I shall refer to several excellent little works 
devoted to the subject ; but I will here attempt to give a 
sketch of the various ways in which a sentence may be 
constructed. 

So often is the copula united to the predicate in 
ordinary language, that the grammarian treats the propo- 
sition as composed of only two parts, the subject and 
predicate, or verb. Thus the proposition, "The sun 
rises," apparently contains nothing but a subject "the 
sun," and a predicate "rises;" but the proposition is 
really equivalent to "the sun is rising," in which the 
copula is distinctly shown. We shall, therefore, con- 
sider the verb or grammatical predicate as containing both 
copula and logical predicate. In Latin one single word 
may combine all the three parts of the proposition, as in 
sum^ " I am ;" and the celebrated exclamation of Csesar 
Veni^ vicN, vici^ " I came, I saw, I conquered," contains 
three distinct and complete propositions in three words. 
These peculiar cases only arise, however, from the parts 
>f the proposition having been blended together and dis* 



LESS. XI.]. ANALYSIS OF SENTENCES. 89 

g^uised in one word ; and in the Latin sum, the letter m 
is a relic of the pronoun me, which is the real subject of 
the proposition. If we had a perfect acquaintance with 
the Grammar of any language it would probably not con- 
tradict the logical view of a sentence, but would perhaps 
explain how the several parts of the complete proposition 
had become blended and apparently lost, just as the 
words will and not are blended in the colloquial " I wont.'' 

A grammatical sentence may contain any number of 
distinct propositions, which admit of being separated but 
which are combined together for the sake of brevity. In 
the sentence, 

"Art is long and Time is fleeting," 
there are two distinct subjects, Art and Time, and two 
predicates, "long" and "fleeting," so that we have simply 
two propositions connected by the conjunction and. We 
may have however several distinct subjects with one and 
the same predicate ; as in 

" Thirty days hath September, 
April, June, and November. " 

In this well-known couplet the predicate " having 
thirty days " is placed first for the sake of emphasis, and 
there are four subjects, September, April, &c., of each of 
which it is affirmed. Hence these lines really contain four 
distinct propositions. 

Again, there maybe one subject with a plurality of 
predicates, so that several different propositions are as- 
serted without the repetition of the subject and copula. 
Thus the sentence 

"Nitrogen is a colourless, tasteless, inodorcus gas, 
slightly lighter than air," contains one subject only, Nt- 
trogen, but four or five predicates ; it is plainly equiva- 
lent to "Nitrogen is colourless," "Nitrogen is tasteless," 
" Nitrogen is a gas," and so on. 

Lastly, we may have several subjects and severa) 



go LOGICAL ANALYSIS [less. 

predicates all combined in the same sentence, and with 
only one copula, so that each predicate is asserted of 
each subject ; and a great number of distinct propositions 
are condensed into one brief sentence. Thus in the sen- 
tence, "Iron, Copper, Lead and Zinc are abundant, cheap 
and useful metalsj" we have evidently four subjects, and 
we may be said to have four predicates, "abundant," 
"cheap," "useful," and "metal" As there is nothing to 
prevent our applying each predicate to each subject the 
sentence really contains l6 distinct propositions in only 
II words; thus "Iron is abundant," "'Iron is cheap," 
"Copper is abundant," "Copper is cheap," and soon. 
In the curious sentence, — 

*' Hearts, tongues, figures, scribes, bards, poets, can- 
not think, speak, cast, write, sing, number, his love to 
Antony"^," Shakspeare has united six subjects and six 
predicates, or verbs, so that there are, strictly speaking, 
six times six or thirty-six propositions. 

In all the cases above noticed the sentence is said to 
be compound, and the distmct propositions combined 
together are said to be coordinate with each other, that is 
of the same order or kind, because they do not depend 
upon each other, or in any way affect each other's truth. 
The abundance, cheapness, or utility of iron need not 
be stated in the same sentence with the qualities of cop- 
per, lead or zinc ; but as the predicates happen to be the 
same, considerable trouble in speaking or writing is 
saved by putting as many subjects as possible to the 
same set of predicates. It is truly said that brevity 
is the soul of wit, and one of the great arts of compo- 
sition consists in condensing as many statements as 
possible into the fewest words, so long as the meaning i? 
not confused thereby. 

• Antony and Cleopatra, Act III. So. i. 



XL] OF SENTENCES. 91 

Propositions are however combined in a totally dif- 
ferent manner when one proposition forms a part of the 
subject or predicate of the other. Thus in the sen- 
tence, "The man who is upright need not fear accusa- 
tion," there are two verbs, and two propositions, but one 
of these only describes the subject of the other; "who 
is upright " evidently restricts the application of the pre- 
dicate " need not fear accusation " to a part of the class 
"man." The meaning of the whole sentence might be 
expressed in the form 

" The upright man need not fear accusation. " 
And it is clearly seen that the clause or apparent propo- 
sition is substituted for an adjective. Such a clause or 
proposition is called subordinate, because it merely as- 
sists in the formation of the principal sentence, and has 
no meaning apart from it ; and any sentence containing 
a subordinate clause is said to be complex. Almost any , 
part of a sentence may thus be replaced by a subordinate 
clause. Thus in "Oxygen and Nitrogen are the gases 
which form the largest part of the atmosphere," there is a 
subordinate clause making part of the predicate, and the 
meaning might be expressed nearly as well in this way, 
" Oxygen and Nitrogen are the gases forming the largest 
part of the atmosphere." 

In the case of a modal proposition (see p. 69), or one 
which states the manner in which the predicate belongs 
to the subject, the mode may be expressed either by an 
adverb, or by a subordinate clause. "As a man lives so 
he dies" is such a proposition; for it means, "a man 
dies as he lives," and " as he lives " is equivalent to an 
adverb ; if he lives well, he dies well ; if he lives badly, 
he dies badly. Adverbs or adverbial clauses may alsc 
specify the time, place, or any other circumstance con- 
cerned in the truth of the main proposition. 

Assuming the reader to be acquainted with the gram- 



92 LOGICAL ANALYSIS [less, 

matical terms used, we may thus state the parts of which 
the most complex sentence must consist. 

The subject may consist of — 

1. A noun ; as in " The Queen reigns." 

2. A pronoun ; as in ^'She reigns." 

3. An adjective converted into a noun ; as in " Whites 
are civilized." 

4- A gerund ; as " Seeing is believing." 

5. An infinitive ; as " To see is to believe." 

6. A subordinate clause ; as " Who falls from virtue 
is lost." 

The subject may be qualified or restricted by combin- 
ing with it an attribute which may be expressed in any of 
the following ways : 

1. An adjective; as, ^^ Fresh air is wholesome." 

2. A participle ; as " Falling stars are often seen.'* 

3. A noun used as an adjective ; as " Iron ships are 
now much employed." 

4. A noun and preposition ; as "ships of iron are now 
much employed." 

5. A possessive case ; as " Chatha^n^s son was the 
great minister Pitt." 

6. A noun in apposition ; as " The Metropolis Londo?ir 
is the most populous of cities." 

7. A gerund or dative infinitive ; as, " The desire to ^0 
abroad is common in Englishmen." 

The predicate consists almost always of a verb, which 
often has some object or qualifying words; thus it may 
be-- 

1. A simple tense of a complete verb ; as " The sun 
risesP 

2. A compound tense ; as " The sun has riscn^ 

3. An incomplete verb and complement; as "THa?. 
sea appears roughP 



XL] OF SENTENCES, 93 

4. The verb " to be" and an adjective : as " Time is 
fteetingP 

5. A verb with an object ; as " Warmth melts ice.^^ 

6. A verb with an adverbial; as "The snow falls 
thickly:' 

The object of a verb is usually a noun or pronoun, 
but any other of the six kinds of expressions which may 
serve as a subject may also serve as an object. 

The adverbial qualifying a verb and expressing the 
manner, time, place, or other circumstance affecting the 
proposition may be — 

1. An adverb; as "The days pass slowly^' 

2. A noun and preposition; as "The resolution was 
passed by a large majority P 

3. An absolute phrase ; as " The snow melts, the sun 
having risen,'' 

4. A dative infinitive ; as " She stoops to conquer:' 

5. Any phrase equivalent to an adverb ; as " The divi- 
dends are paid twice a year:' 

Various modes of exhibiting the construction of sen- 
tences by symbols and names for the several parts have 
been invented ; but I believe that by far the simplest and 
most efficient mode is to exhibit the construction in the 
form of a diagram. Any two or more parts of a sentence 
which are co-ordinate with each other, or bear the same 
relation to any other part, are written alongside each 
other, and coupled together by a bracket ; thus the dia- 
gram,— 

Iron \ ( abundant, 

Copper I I cheap, 

Lead j ^^^ I useful 

Zinc 3 I metals, 

clearly shows that there are four co-ordinate subjects, 



94 LOGICAL ANALYSIS [LESS. 

and four co-ordinate predicates in the example previously 
taken. 

Whenever one part of a sentence is subordinate to 
another part it may be connected with it by a line drawn 
in any convenient direction. Thus the analysis of the 
following sentence is readily shown by the diagram below 
it :— 

" No one who is a lover of money, a lover of pleasure, 
and a lover of glory, is likewise a lover of mankind ; but 
only he who is a lover of virtue.'' 

/ a lover of money, 
who is <[ a lover of pleasure, 

I I a lover of glory, 

one is not 



, . , a lover of mankind, 
he only is * 



who is a lover of virtue. 

We see that the sentence is both compound and com- 
plex, that is to say it contains two principal coordinate 
propositions with a common predicate, " a lover of man- 
kind." The first proposition is negative and its subject is 
described by three subordinate clauses, while the second 
proposition is affirmative and has one subordinate clause. 

I conclude this somewhat lengthy lesson with the 
analysis of a few sentences, of which the first consists 
of some remarkably complex lines from a poem of Bur- 
bidge : 

" He who metes, as we should mete, 
Could we His insight use, shall most approve. 
Not that which fills most space in earthly eyes, 
But what — though Time scarce note it as he flies^— 
Fills, like this little daisy at my feet, 
Its function best of diligence in love." 



XL] OF SENTENCES, ^ 

which fills most space in earthly eyes 

I , 

TT 1 Ti \ ^ot that 

He shall most approve | ^^^ ^^^^ ^^^^ ^^^^ 



who metes its function of like this little 

as'^^ould mete diligence in daisy at my 

love feet, 



could we His insis^ht use. 'T 



-Y— 



.. J 



though Time scarce note it 

I 

as he flies. 

" Most sweet it is with unuplifted eyes 

To pace the ground, if path there be or none, 
While a fair region round the traveller lies 

Which he forbears again to look upon ; 
Pleased rather with some soft ideal scene, 
The work of fancy, or some happy tone 
Of meditation slipping in between. 
The beauty coming, and the beauty gone." 

Wordsworth. 
It is most sweet - 
! 

To pace the ground 

/ -> ; = — , 

with unuplifted if path while a fair region 

eyes th e J ^^ round the 

or none traveller li-es 



^'hich (region) he (the traveller) forbears to look upon 

I some soft ideal scene 
^"■^ — 77 ' 
the work of fancy 
or some happy tone of meditation 



slipping in between the beauty coming 
and the beauty gone. 

In the above sentence there is evidently one Fubjec* 



06 



LOGICAL ANALYSIS 



[LESS. 



" to pace the ground," which by means of the pronoun //, 
is connected with the predicate most sweet. The main 
part of the sentence however consists of three adverbials, 
expressing the manner and surrounding circumstances, 
and the third adverbial is developed in a very complicated 
manner. The sentence is not compound, but is complex 
on account of four subordinate propositions. 

In the following sentence there is strictly but one 
principal proposition, " We find," but this is only a mode 
of introducing the true purport of the sentence, " the two 
classes of intellectual operations have much that is differ- 
ent, much that is common." 

" When the notions with which men are conversant in 
the common course of life, which give meaning to their 
familiar language and which give employment to their 
hourly thoughts, are compared with the ideas on which 
exact science is founded, we find, that the two classes of 
intellectual operations have much that is different, much 
ihat is common.'* 
we find — that the two classes (* f) 

of intellectual i much that is different 
operations have ( much that is common 



When the notions 

A_ 



•«. 



are compared 



which give which give 
employ- 
ment to 



with the ideas f 



meanmg 
to their 
familiar 
language 



their hourly 
thoughts 



on which 

.exact science is 

founded. 



with which 
men are 
conversant 
in the 
common 
course 
of hfe 

Here the two classes form a collective term, and have 
two coordinate predicates rendering the sentence so far a 
compound one. The greater part of the sentence, how- 
ever, consists of a complicated subordinate sentence of 



XL] 



OF SENTENCES. 



97 



the nature of an adverbial, expressing the time or occa- 
sion when this is found to be the case. 

As a last example we take the sentence given below:— 
" The law of gravitation, ^v.q iitost universal truth at 
which human reason has yet arrived, expresses not merely 
the general fact of the mutual attraction of all matter ; not 
merely the vague statement that its influence decreases as 
the distance increases, but the exact numerical rate at 
which that decrease takes place ; so that when its amount 
is known at any one distance it may be exactly calculated 
for any other." 

at which human reason has yet arrived 



the most universal truth 



The law of gravitation expresses 



not merely the 
general fact 



not merely the 
vague stat(^ment 



of the mutual 
attraction of all 
matter 



that its influence 
decreases 



as the distance 
increases 



but the exact 
numerical rate 



at which that 
decrease takes 
place 



so that its amount may be calculated for any other dis< 

I [tance 

when it is known at any one distance. 

W. S. Dalgleish's Grammatical Analysis^ or 
J. D. Morell's Analysis of Sentences. 
Alex. Bain's English Composition and Rhe- 
toric^ pp. 91 — 117, treats of construction of 
sentences. 



LESSON XII. 

THE PREDICABLES, DIVISION, AND 
DEFINITION. 

It is desirable that the reader, before proceeding further, 
should acquire an exact comprehension of the meaning of 
certain logical terms which are known as the Predicables, 
meaning the kinds of terms or attributes which can always 
be predicated of any subject. These terms are five in 
number ; genus, species, difference, property, and acci- 
dent ; and when properly employed are of exceeding use 
and importance in logical science. It would neither be 
possible nor desirable in this work to attempt to give any 
idea of the various and subtle meanings which have been 
attributed to the predicables by ancient writers, and the 
most simple and useful view of the subject is what alone 
can be given here. 

Any class of things may be called a genus (Greek 
yevos, race or kind), if it be regarded as made up of two 
or more species. " Element" is a genus when we con- 
sider it as divided into the two species "metallic and 
non-metallic." Triangle is a genus as regards the species 
acute-angled, right-angled, and obtuse-angled. 

On the other hand, a species is any class which is re- 
garded as forming part of the next larger class, so that 
the terms genus and species are relative to each otherj 
the genus being the larger class which is divided, and the 
species the two or more smaller classes into which the 
genus is divided. 

It is indispensable, however, to regard these expres- 
sions in the double meaning of extension and intension. 



LESS. XII ] THE PREDICABLES, ETC. 9«p 

From the explanation of these different meanings in 
Lesson V. it will be apparent that the extent of a genua 
or species is simply the number of individuals included 
in it, and there will always be fewer individuals in the 
species than in the genus. In extent the genus book in- 
cludes all books of whatever size, language, or contents ; 
if divided in respect to size the species of book are folio, 
quarto, octavo, duodecimo, &c. ; and, of course, each of 
these species contains much fewer individual books than 
the whole genus. 

In intension the genus means, not the individual 
things contained in it, but the sum of the qualities com- 
mon to all those things, and sufficient to mark them out 
clearly from other classes. The species similarly means 
the sum of the qualities comm.on to all the individuals 
forming part of the genus, and sufficient to mark them out 
from the rest of the genus, as well as from all other things. 
It is evident, therefore, that there must be more qualities 
implied in the meaning of the species than of the genus, 
for the species must contain all the qualities of the genus, 
as well as a certain additional quality or qualities by 
which the several species are distinguished from each 
other. Now these additional qualities form the difference, 
which may be defined as the quality or sum of qualities 
which mark out one part of a genus from the other part or 
parts. The difference (Latin diffei^entia, Greek hia- 
(popd) cannot have any meaning except in intension; 
and when we use all the terms wholly in intension we may 
say that ^/le diffe^^ence added to the genus makes the species. 
Thus if ^'building" be the genus, and we add the differ- 
ence " used for a dwelling," we get the species " house.'- 
If we take "triangle" as the genus, it means the sum of 
the qualities of '^ three-sided rectilineal figure ;" if we add 
the quality of "having two sides equal," we obtain the 
species "isosceles triangle." 

7—2 



loo THE PREDICABLES, DIVISION, [less. 

It will easily be seen that the same class of things 
may be both a genus and a species at the same time, ac- 
cording as we regard it as divided into smaller classes or 
forming part of a larger class. Thus triangle, which is 
a genus as regards isosceles triangle, is a species as re- 
gards right-lined geometrical figures. House is a species 
of building, but a. genus with respect to mansion, cottage, 
villa, or other kinds of houses. We may, in fact, have an 
almost interminable chain of genera and species, each 
class being a species of the class next above it, and a 
genus as regards that next below. Thus the genus Bri- 
tish subject has the species Born in the United Kingdom, 
Colonial-born, and Naturalised. Each of these becomes 
a genus as regards the species male and female; each 
species again may be divided into adult and minor, edu- 
cated, uneducated, employed in some occupation or un- 
employed, self-maintaining, maintained by friends, or 
pauper ; and so on. The subdivision may thus proceed 
until we reach a class of so restricted extent, that it 
cannot be divided except into individuals ; in this case 
the species is called the lowest species or infima species. 
All the intermediate genera and species of the chain are 
called subaltern (Latin sub^ under, and alter^ the other of 
two), because they stand one under the other. If there be 
a genus which is not regarded as a species, that is as 
part of any higher genus, it is called the summum genus, 
the highest genus, or geiius generalissimum^ the most 
general genus. It is questionable whether we can thus 
set any limit to the chain of classes. The class British 
subject is certainly not an absolute suminimi genus, 
since it is but a species of 7nan^ which is a species of 
animal, living being, inhabitant of the earth, substance, 
and so on. If there were any real su7nmuin genus it 
would probably be " Being,'' or " Thing,'' or *' Object con- 
ceivable ;" but we may usefully employ the term to signify 



xir.] AND DEFINITION, loi 

the highest class of things comprehended in any science 
or classification. Thus "material substance" is the sum- 
mum genus examined in the science of chemistry; "in- 
habitant of the United Kingdom" is the summum genus 
enumerated and classified in the British census. Logi- 
cal terms are only a species of words or phrases, but they 
are the summum genus as regards logic, which has no- 
thing to do with the various parts of speech and the 
relations of words, syllables, and letters, examined by 
grammarians. 

Several very useful expressions have been derived 
from the words genus and species. When a thing is 
so peculiar and unlike other things that it cannot easily 
be brought into one class with them, it is said to be sui 
generis, or of its own genus ; thus the rings of Saturn are 
so different from anything else among the heavenly bodies 
that they may. fairly be called sui generis. In zoology, 
the Ornithorhynchus, or Australian Duck-bill, the Amphi- 
oxus, and some other animals, are so peculiar that they 
may be called sui generis. When a substance is the 
same in all its parts, or when a number of things are all 
alike, we say that they are homogeneotis (Greek oyios, like, 
yepos, kind), that is of the same nature ; otherwise they 
may be called heterogetieous (Greek crepos, other). 

It is necessary to distinguish carefully the purely lo- 
gical use of the terms genus and species from their pecu- 
liar use in natural history. A species is there a class 
of plants and animals supposed to have descended from 
common parents, and to be the narrowest class possessing 
a fixed form ; the genus is the next higher class. But if 
we accept Darwin's theoiy of the origin of species, this 
definition of species becomes entirely illusory, since dif- 
ferent genera and species must have according to this 
theory descended from common parents. The species 
then denotes a merely arbitrary amount of resemblance 






I02 THE PREDICABLES, DIVISION, [less. 

which naturahsts choose to fix upon, and which it is not 
possible to define more exactly. This use of the term, 
then, has no connection whatever with the logical use, 
according to which any class of things whatever is a 
species, provided it is regarded as part of a wider class or 
genus. 

The fourth of the Predicables is Property (Latin pro- 
prium, Greek Ihiov, own), which it is hardly possible to 
define in a manner free from objection and difficulty, but 
which may perhaps be best described as any quality 
which is common to the whole of a class, but is not neces- 
sary to mark out that class from other classes. Thus it is 
a property of the genus '* triangle" to have the three in- 
ternal angles equal to two right angles ; this is a very 
remarkable circumstance, which is always true of tri- 
angles, but it is not made a part of the genus, or is not 
employed in defining a triangle, because the possession of 
three straight sides is a sufficient mark. The properties of 
geometrical figures are very numerous ; the Second Book 
of Euclid is occupied in proving a few properties of rect- 
angles ; the Third Book similarly of circles. As we com- 
monly use the term property it may or may not belong to 
other objects as well as those in question; some of the 
properties of the circle may belong also to the ellipse ; 
some of the properties of man, as for instance the power 
of memory, or of anger, may belong to other animals. 

Logicians have invented various subtle divisions of pro- 
perties, but it will be sufftcient to say that 2^ peculiar pro- 
perty is one which belongs to the whole of a class, and to 
that class only, as laughter is supposed to belong only to 
mankind ; the property of containing the greatest space in 
a line of given length is peculiar to circles. When a pro- 
perty is not peculiar, it may belong to other classes of 
objects •as well as that of which it is called the property. 
We may further distinguish the Generic Property, or that 



XII.] AND DEFINITION 103 

which belongs to the whole of the genus, from the 
Specific Property, which belongs to the whole of a lowest 
species. 

Lastly, an accident (Latin accidens^ Greek cru/x/Se/S?;- 
Koi) is any quality which may indifferently belong or 
not belong to a class, as the case may be, without 
affecting the other qualities of the class. The word 
means that which y^Z/y or happens by chance, and has no 
necessary connection with the nature of a thing. Thus 
the absolute size of a triangle is a pure accident as 
regards its geometrical properties ; foi; whether the side 
of a triangle be ^^ of an inch or a million miles, what- 
ever Euclid proves to be true of one is true of the other. 
The birthplace of a man is an accident concerning him, as 
are also the clothes in which he is dressed, the position in 
which he rests, and so on. Some writers distinguish se- 
parable and inseparable accidents. Thus the clothes in 
which a man is dressed is a separable accident, because 
they can be changed, as can also his position, and many 
other circumstances ; but his birthplace, his height, his 
Christian name, &c., are inseparable accidents, because 
they can never be changed, although they have no neces- 
sary or important relation to his general character. 

As an illustration of some part of the scheme of clas- 
sification described under the name of Predicables, I may 
here give, as is usual in manuals of Logic, the Tree of 
Porphyry, a sort of example of classification invented by 
one of the earliest Greek logicians, named Porphyrius. . 
I have simplified the common form in which it is given 
by translating the Latin names and omitting superfluous 
words. 

In this Tree we observe a succession of genera and 
species — Substance, Body, Living Being, Anim^al and 
Man. Of these Substance is Xh^ sitmniiim genus ^ because 
it is not regarded as a species of any higher class ; Man 



I04 THE PREDICABLES, DI VIST 01/^ [less. 

is the injima species^ because it is a class not divided in- 
to any lower class, but only into individuals, of whom it is 

Substance, 



Corporeal, 



Incorporeal, 



Body, 



Animate, 



Inanimate, 



Living Being, 



Sensible, 



Insensible, 



Animal, 



Rational, 



Irrational, 



Man, 



Socrates, 



Plato, 



and others. 



usual to specify Socrates and Plato. Body, Living Being, 
and Animal are called subaltern genera and species, be- 
cause each is a species as regards the next higher genus, 
and a genus as regards the next lower species. The 
qualities implied in the adjectives Corporeal, Animate, 
Sensible {i.e. capable of feeling) and Rational are the 
successive differences which occasion a division of each 
genus into species. It will be evident that the negative 
parts of the genera, namely Incorporeal Substance, In- 



XII.] AND DEFINITION, 105 

animate Body, &c., are capable of subdivision, which has 
not been carried out in order to avoid confusing the 
figure. 

Logical division is the name of the process by which 
we distinguish the species of which a genus is composed. 
Thus we are said to divide the genus " book " when we 
consider it as made up of the groups foHo, quarto, octavo, 
duodecimo books, &c., and the size of the books is in this 
case the ground, basics, or principle of division, commonly 
called the Fundamentum Divisionis. In order that a quality 
or circumstance may be taken as the basis of division, it 
must be present with some and absent with others, or 
must vary with the different species comprehended in the 
genus. A generic property of course, being present in the 
whole of the genus, cannot serve for the purpose of divi- 
sion. Three rules may be laid down to which a sound 
and useful division must conform : 

1. The constituent species must exclude each other. 

2. The constituent species must be equal when add- 
ed together to the genus. 

3. The division must be founded upon one principle 
or basis. 

It would be obviously absurd to divide books into 
folio, quarto, French, German and dictionaries, because 
these species overlap each other, and there may be French 
or German dictionaries which happen to be quarto or 
folio and belong to three different species at once. A 
division of this kind is said to be a Cross Division, because 
there is more than one principle of division, and the seve- 
ral species in consequence cross each other and produce 
confusion. If I were to divide rectilineal figures into tri- 
angles, parallelograms, rectangles and polygons of more 
than four sides, I should commit all the possible faults in 
one division. The species parallelogram and rectangle 
do not exclude each other, since all rectangles must be 



io6 THE PREDICABLES, DIVISION, [less. 

parallelograms ; the constituent species are not altogether 
equal to the genus rectilineal figure, since irregular four- 
sided figures which are not parallelograms have been 
omitted ; and there are three principles of division, namely 
the number of sides, the directions of those sides, and the 
angles contained. But when subdivision is employed, 
and each of the species is considered as a genus which 
may be subjected to a further separation, a new principle 
of division may and in fact must be employed each time. 
Thus I can divide rectilineal figures according to the three 
principles mentioned above : 

Rectilineal Figure 

r ' -1 

3 sides 4 sides more than 4 sides 

Triangle Quadrilateral PolygOQ 

J 1 , 

with parallel sides without parallel 

Parallelogram sides 

Trapezium. 

Here the principles of division are the number of their 
sides, and in the case of four-sided figures their paral- 
lelism. Triangles do not admit of division in this second 
respect. We may make a new division of parallelograms, 
adopting the equality of sides and the size of the angles 
as the principles ; thus : 

Parallelogram 



, I ; ; 1 

adjoining sides adjoining sides 

equal not equal 



I 1 



right- not right- right- not right- 

angled angled angled angled 

Square Rhombus Oblong Rhomboid. 

The most perfect divisions in a logical point of view 
are produced by continually dividing each genus into two 



XII.] AND DEFINITION, 107 

species by a difference, of which an example has been 
given in the Tree of Porphyry. This process is called 
Dichotomy (Greek hix^i^ in two; re/xvo), to cut); it is also 
called Exhaustive Division because it always of necessity 
obeys the second rule, and provides a place for, every 
possible existing thing. By a Law of Thought to be con- 
sidered in the next Lesson, every thing must either have 
a quality or not have it, so that it must fall into one or 
other division of the genus. This process of exhaustive 
division will be shewn to have considerable importance in 
Lesson XXII I., but in practice it is not by any means 
always necessary or convenient. It would, for instance, 
produce a needlessly long classification if we divided rec- 
tilineal figures thus : 

Rectilineal figure 

. r ^ : 

3-sided not 3-sided 



Triangle 



, — , 

4-sided not 4-sided 

Quadrilateral 



'I 



5-sided not 5-sided 

Pentagon &c. 

As we know beyond all doubt that every figure must 
have 3, 4, 5, 6, or more sides, and no figure can belong to 
more than one group, it is much better at once to enume- 
rate the parts as Triangle, Quadrilateral, Pentagon, Hexa- 
gon, &:c. Again, it would be very awkward if we. divided 
the counties of England into Middlesex and not-Middle- 
sex; the latter into Surrey and not- Surrey; the latter, 
again, into Kent and not-Kent. Dichotomy is useless, 
and even seems absurd in these cases, because we can 
observe the rules of division certainly in a much briefer 
division. But in less certain branches of knowledge our 
divisions can never be free from possible oversight unless 
they proceed by dichotomy. Thus, if we divide the popula- 
tion of the world into three branches, Aryan, Semitic, and 



io8 THE PREDICABLES, DIVISION, [less. 

Turanian, some race might ultimately be discovered which 
is distinct from any of these, and for which no place has 
been provided ; but had we proceeded thus — 

Man 

-J . 



Aryan not- Aryan 



Semitic not-Semitic 

L 



f— 



Turanian not-Turanian, 

it is evident that the new race would fall into the last 
group, which is neither Aryan, Semitic, nor Turanian. All 
the divisions of naturalists are liable to this inconvenience. 
If we divide Vertebrate Animals into Mammalia, Birds^ 
Reptiles, and Fish, it may any time happen that a new 
form is discovered which belongs to none of these, and 
therefore upsets the division. 

A further precaution required in Division is not to 
proceed from a high or wide genus at once to a low 
or narrow species, or, as the phrase is, divisio non facial 
salturn (the division should not make a leap). The 
species should always be those of the proximate or next 
higher genus ; thus it would obviously be inconvenient to 
begin by dividing geometrical figures into those which 
have parallel sides and those which have not; but this 
principle of division is very proper when applied to the 
proximate genus. 

Logical division must not be confused with physical 
division or Partition, by which an individual object, as a 
tree, is regarded as composed of its separate parts, root, 
trunk, branches, leaves, &e. There is even a third and 
distinct process, called Metaphysical Division, which con- 
sists in regarding a thing as an aggregate of qualities, 
and separating these in thought ; as when we discriminate 
the form, colour, taste, and smell of an orange. 

Closely connected with the subject of this Lesson is 



XII.] AND DEFINITION, 109 

the process of Logical Definition, by which we determine 
the common quahties or marks of the objects belonging 
to any given class of objects. We must give in a defini- 
tion the briefest possible statement of such qualities as 
are sufficient to distinguish the class from other classes, 
and determine its position in the general classification of 
conceptions. Now this will be fulfilled by regarding the 
class as a species, and giving the proximate genus and 
the difference. The word genus is here used in its inten- 
sive meaning, and denotes the qualities belonging to all 
of the genus, and sufficient to mark them out ; and as the 
difference marks out the part of the genus in question, 
we get a perfect definition of the species desired. But we 
should be careful to give in a definition no superfluous 
marks ; if these are accidents and do not belong to the 
whole, the definition will be improperly narrowed, as if 
we were to define Quadrilateral Figures as figures with 
four equal sides; if the superfluous marks belong to all 
the things defined they are Properties^ and have no effect 
upon the definition whatever. Thus if I define parallelo- 
grams as " four- sided rectilineal figures, with the opposite 
sides equal and parallel, and the opposite angles equal," 
I have added two properties, the equality of the opposite 
sides and angles which necessarily follow from the paral- 
lelism of the sides, and only add to the complexity of the 
definition without rendering it more precise. 

There are certain rules usually given in logical works 
which express the precautions necessary in definition. 

1. A definition should state the essential attributes of 
the species defined. So far as any exact meaning can be 
given to the expression "essential attributes," it means, 
as explained above, the proximate genus and difference. 

2. A definition must not contain the na^ne defined. 
For the purpose of the definition is to make the species 
known, and as long as it is not known it cannot sei've to 



no THE PREDICABLES, DIVISION, [less 

make itself known. When this rule is not observed, there 
is said to be ^ circulus in definiendoj or ^ a circle in defin- 
ing/ because the definition brings us round again to the 
very word from which we started. This fault will usually 
be committed by using a word in the definition which is 
really a synonym of the name defined, as if I were to 
define " Plant" as " an organized being possessing vege- 
table life," or elements as simple substances, vegetable 
being really equivalent to plant, and simple to elementary. 
If I were to define metals as "substances possessing me- 
tallic lustre," I should either commit this fault, or use the 
term metallic lustre in a sense which would admit other 
substances, and thub break the following rule. 

3. The deJinitio7i must be exactly equivalent to the 
species deji7ied, that is to say, it must be an expression the 
denotation of which is neither narrower nor wider than 
the species, so as to include exactly the same objects. 
The definition, in short, must denote the species, the 
whole species, and nothing but the species, and this may 
really be considered a description of what a definition is. 

4. A definition must not be expressed in obscure, figura- 
tive or ainbiguous language. In other words, the terms 
employed in the definition must be all exactly known, 
otherwise the purpose of the definition, to make us ac- 
quainted with the "sufficient marks of the species, is 
obviously defeated. There is no worse logical fault than 
to define ignotum per ignotius, the unknown by the still 
more unknown. Aristotle's definition of the soul as * The 
Entelechy, or first form of an organized body which has 
potential life,' certainly seems subject to this objection. 

5. And lastly,^ definition must not be negative where 
it can be affirmative. This rule however is often not 
applicable, and is by no means always binding. 

Read Mr Mill on the nature of Classification and the 



XII.] AND DEFINITION. lU 

five Predicables, System of Logic ^ Book I. Chap, 
VII. For ancient Scholastic Views concerning De- 
finition, see Mansel's Artis Logicce Rudimenta. 
(Aldrich), App. Note C. 



LESSON XIII. 

PASCAL AND DESCARTES ON METHOD. 

It may be doubted whether any man ever possessed a 
more acute and perfect intellect than that of Blaise 
Pascal. He was born in 1623, at Clermont in Auvergne, 
and from his earliest years displayed signs of a remark- 
able character. His father attempted at first to prevent 
his studying geometry, but such was Pascal's genius and 
love of this science, that, by the age of twelve, he had 
found out many of the propositions of Euclid's first book 
without the aid of any person or treatise. It is difficult 
to say whether he is most to be admired for his mathe- 
matical discoveries, his invention of the first calculating 
machine, his wonderful Provincial Letters written against 
the Jesuits, or for his profound Pensees or Thoughts, a 
collection of his reflections on scientific and religious 
topics. 

Among these Thoughts is to be found a remarkable 
fragment upon Logical method, the substance of which is 
also given in the Port Royal Logic, It forms the second 
article of the Pe7tsees, and is entitled Reflexions siir la 
Geometrie en general. As I know no composition in 
which perfection of truth and clearness of expression are 
more nearly attained, I propose to give in this lesson a 
free translation of the more important parts of this 



112 PASCAL AND DESCARTES [less 

fragment, appending to it rules of method from the 
Port Royal Logic, and from Descartes' celebrated Essay 
on Method, The words of Pascal are nearly as follows. 

"The true method, which would furnish demonstra- 
tions of the highest excellence, if it were possible to 
employ the method fully, consists in observing two prin- 
cipal rules. The first rule is not to employ any term of 
which we have not clearly explained the meaning; the 
second rule is never to put forward any proposition which 
we cannot demonstrate by truths already known ; that is 
to say, in a word, to define all the terms ^ and to prove all 
the propositions. But, in order that I may observe the 
rules of the method which I am explaining, it is neces- 
sary that I declare what is to be understood by Definition, 

"We recognise in Geometry only those definitions 
which logicians call Nominal Definitions, that is to say, 
only those definitions which impose a name upon things 
clearly designated in terms perfectly known ; and I speak 
only of those definitions." 

Their value and use is to clear and abbreviate dis- 
course by "expressing in the single name which we 
impose what could not be otherwise expressed but in 
several words ; provided nevertheless that the name im- 
posed remain divested of any other meaning which it 
might possess, so as to bear that alone for which we 
intend it to stand. 

" For example, if we need to distinguish among 
numbers those which are divisible into two equal parts, 
from those which are not so divisible, in order to avoid 
the frequent repetition of this distinction, we give a name 
to it in this manner : — we call every number divisible into 
two equal parts an Even Number, 

" This is a geometrical definition, because after having 
clearly designated a thing, namely any number divisible 
into two equal parts, we give it a name divested of every 



XIII.] ON METHOD. 113 

other meaning, which it might have, in order to bestow 
upon it the meaning designated. 

" Hence it appears that definitions are very free, and 
that they can never be subject to contradiction, for there 
is nothing more allowable, than to give any name we wish 
to a thing which we have clearly pointed out. It is only 
necessary to take care that we do not abuse this liberty of 
imposing names, by giving the same name to two differ- 
ent things. Even that would be allowable, provided that 
we did not confuse the results, and extend them from 
one to the other. But if we fall into this vice, we have a 
very sure and infallible remedy ; — it is, to substitute men- 
tally the definition in place of the thing defined, and to 
hold the definition always so present in the mind, that 
every time we speak, for instance, of an even number, we 
may understand precisely that it is a number divisible 
into two equal parts, and so that these two things should 
be so combined and inseparable in thought, that as often 
as one is expressed in discourse, the mind may direct it- 
self immediately to the other. 

" For geometers and all who proceed methodically 
only impose names upon things in order to abbreviate 
discourse, and not to lessen or change the ideas of the 
things concerning which they discourse. They pretend 
that the mind always supplies the entire definition of the 
brief terms which they employ simply to avoid the con- 
fusion produced by a multitude of words. 

" N othing prevents more promptly and effectively the 
insidious fallacies of the sophists than this method, which 
we should always employ, and which alone suffices to 
banish all sorts of difficulties and equivocations. 

" These things being well understood, I return to my 
explanation of the true method, which consists, as I said, 
in defining everything and proving everything. 

*^ Certainly this method would be an excellent one, 

8 



TI4 ^ PASCAL AND DESCARTES [less 

were it not absolutely impossible. It is evident that the 
first terms we wished to define would require previous 
terms to serve for their explanation, and similarly the 
first propositions we wished to prove, would presuppose 
other propositions preceding them in our knowledge ; and 
thus it is clear that we should never arrive at the first 
terms or first propositions. 

"Accordingly in pushing our researches further and 
further, we arrive necessarily at primitive words which we 
cannot define, and at principles so clear, that we cannot 
find any principles more clear to prove them by. Thus 
it appears that men are naturally and inevitably incapa- 
ble of treating any science whatever in a perfect method \ 
but it does not thence follow that we ought to abandon 
every kind of method... ...The most perfect method avail- 
able to men consists not in defining everything and de- 
monstrating everything, nor in defining nothing and de- 
monstrating nothing, but in pursuing the middle course 
of not defining things which are clear and understood by 
all persons, but of defining all others ; and of not proving 
truths known to all persons, but of proving all others. 
From this method they equally err who undertake to de- 
fine and prove everything, and they who neglect to do it 
in things which are not self-evident." 

It is made plain in this admirable passage that we 
can never by using words avoid an ultimate appeal to 
things, because each definition of a word must require 
one or more other words, which also will require defini- 
tion, and so on ad tnjimiujn. Nor must we ever return 
back upon the words already defined ; for if we define A 
by j5, and B by C, and C by Z>, and then D by ^, we 
commit what may be called a circulus in defiiiiendoj^ a 
most serious fallacy, which might lead us to suppose tha.t 
we know the nature of ^, B^ C, and Z), when we really 
know nothing about them. 



Xiii.] ON METHOD, 115 

Pascal's views of the geometrical method were clearly 
summed up in the following rules, inserted by him in the 
Port Royal Logzc"^. 

1. To admit no terms in the least obscure or equivo- 
cal without defining them. 

2. To employ in the definitions only terms perfectly 
known or already explained. 

3. To demand as axioms only truths perfectly evi- 
dent. 

4. To prove all propositions which are at all obscure, 
by employing in their proof only the definitions which 
have preceded, or the axioms which have been accorded, 
or the propositions which have been already demonstrated, 
or the construction of the thing itself which is in dispute, 
when there may be any operation to perform. 

5. Never to abuse the equivocation of terms by failing 
to substitute for them, mentally, the definitions which 
restrict and explain them. 

The reader will easily see that these rules are much 
more easy to lay down than to observe, since even geo- 
meteK ""re not agreed as to the simplest axioms to assume, 
or the host definitions to make. There are many differ- 
ent opinions as to the true definition of parallel lines, and 
the simplest assumptions concerning their nature; and 
how much greater must be the difficulty of observing 
Pascal's rules with confidence in less certain branches of 
science. Next after Geometry, Mechanics is perhaps the 
most perfect science, yet the best authorities have been 
far from agreeing as to the exact definitions of such 
notions 2,^ force ^ inass^ moment^ power, inertia, and the 
most different opinions are still held as to the simplest 
axioms by which the law of the composition of forces may 
be proved. Nevertheless if we steadily bear in mind, in 

* Mr Spencer Baynes' Translation^ p. 317. 

8—2 



no PASCAL AND DEC A RTFS [less 

studying each science, the necessity of defining every term 
as far as possible, and proving each proposition which 
can be proved by a simpler one, we shall do much to clear 
away error and confusion. 

I also vi'ish to give here the rules proposed by the 
celebrated Descartes for guiding the reason in the attain- 
ment of truth. They are as follows : — 

1. Never to accept anything as true, which we do 
not clearly know to be so ; that is to say, carefully to 
avoid haste or prejudice, and to comprise nothing more 
in our judgments than what presents itself so clearly and 
distinctly to the mind that we cannot have any room to 
doubt it. 

2. To divide each difficulty we examine into as many 
parts as possible, or as may be required for resolv- 
ing it. 

3. To conduct our thoughts in an orderly manner, 
commencing with the most simple and easily known 
objects, in order to ascend by degrees to the knowledge 
of the most complex. 

4. To make in every case enumerations so complete, 
and reviews so wide, that we may be sure of omitting 
nothing. 

These rules were first stated by Descartes in his ad- 
mirable Discourse on Method^ in which he gives his reflec- 
tions on the right mode of conducting the reason, and 
searching for truth in any of the sciences. This little 
treatise is easily to be obtained in the original French, and 
has also been translated into English by Mr Veitch*. 
The reader can be strongly advised to study it. Always to 
observe the rules of Descartes and Pascal, or to know 
whether we in every case observe them properly, is im- 

* Published at Edinburgh in 1850- 



Xiii.j ON METHOD, 117 

possible, but it must nevertheless be valuable to know at 
what we ought to aim. 

Read Locke's brief Essay on the Coiduct of the Un- 
derstattding, which contains admirable remarks on 
the acquirement of exact and logical habits of 
thought. 



LESSON XIV. 

THE LAWS OF THOUGHT. 

Before the reader proceeds to the lessons which treat 
of the most common forms of reasoning, known as the 
syllogism, it is desirable that he should give a careful 
attention to the very simple laws of thought on which all 
reasoning must ultimatel) depend. These laws describe 
the very simplest truths, in which all people must agree, 
and which at the same time apply to all notions which 
we can conceive. It is impossible to think correctly and 
avoid evident self-contradiction unless we observe what 
are called the Ttiree Primary Laws oJ rnougrit, which may 
be stated as follows : 

1. The Law of Identity. Waatever is, Is. 

2. The Law of Contradiction Notliiiig can botli be and 

not be. 

3. The Law of Exch\aed Middle. Everything must 

eitlier be or not be. 

Though these laws when thus stated may seem ab- 
surdly obvious, and were ridiculed by Locke and others 
on that account, I have found that students are seldom 
able to see at first their full meaning and importance. 
It will be pointed out in Lesson XXIH. that logicians have 



I 



I 



ii8 THE LAWS OF THOUGHT. Lless. 

overlooked until recent years the very simple way in which 
all arguments may be explained when these self-evident 
laws aie granted ; and it is not too much to say that the 
w^hole of logic will be plain to those who will constantly 
use these laws as the key. 

The first of the laws may be regarded as the best 
definition we can give of identity or sameness. Could 
any one be ignorant of the meaning of the word Identity, 
it would be sufficient to inform him that everjrthing is 
identical witli itself. 

The second law however is the one which requires 
more consideration. Its meaning is that nothing can 
have at the same time and at the same place contra- 
dictory and inconsistent qualities. A piece of paper may 
be blackened in one part, while it is white in other parts; 
or it may be white at one time, and afterwards become 
black; but we cannot conceive that it should be both 
white and black at the same place and time. A door 
after being open may be shut, but it cannot at once be 
shut and open. Water may feel warm to one hand and 
cold to another hand, but it cannot be both warm and 
cold to the same hand. No quality can both be present 
and absent at the same time ; and this seems to be the 
most simple and general truth which we can assert of all 
things. It is the very nature of existence that a thing 
cannot be otherwise than it is ; and it may be safely said 
that all fallacy and error arise from unwittingly reason- 
ing in a way inconsistent with this law. All statements 
or inferences which imply a combination of contradictory 
qualities must be taken as impossible and false, and the 
breaking of this law is the mark of their being false. It 
can easily be shewn that if Iron be a metal, and every 
metal an element. Iron must be an element or it can be 
nothing at all, since it would combine qualities which are 
inconsistent (sec Lesson xxiii). 



XIV.] THE LAWS OF THOUGHT. 119 

The Law of Excluded Middle is much less self-evident 
than either of the two preceding ones, and the reader will 
not perhaps see at the first moment that it is equally 
important and necessary with them. Its meaning may 
be best explained by saying that it is impossible to men- 
tion any thing and any quality or circumstance, without 
allowing that the quality or circumstance either belongs 
to the thing or does not belong. The name of the law 
expresses the fact that there is no third or middle course ; 
the answer must be Yes or No. Let the thing be rock 
and the quality hard; then rock must be either hard or 
not-hard. Gold must be either white or not white; a 
line must be either straight or not straight; an action 
must be either virtuous or not virtuous. Indeed when 
we know nothing of the terms used we may never- 
theless make assertions concerning them in accordance 
with this law. The reader may not know and in fact 
chemists m.ay not really know with certainty, whether 
vafiadiujn is a metal or not a metal, but any one knows 
that it must be one or the other. Some readers may not 
know what a cycloid is or what an isochronous curve is ; 
but they must know that a cycloid is either an isochro- 
nous curve or it is not an isochronous curve. 

This law of excluded middle is not so evident but that 
plausible objections may be suggested to it. Rock, it 
may be urged, is not always either hard or soft, for it may 
be half way between, a little hard and a little soft at the 
same time. This objection points to a distinction which 
is of great logical importance, and when neglected often 
leads to fallacy. The law of excluded middle affirmed 
nothing about hard and so/t^ but only referred to hara 
and not-hard; if the reader chooses to substitute soft for 
not-hard he falls into a serious confusion between oppasite 
terms and contradictory terms. It is quite possible that 
a thmg may be neither hard nor soft, being half way 



I20 THE LAWS OF THOUGHT, [less. 

between ; but in that case it cannot be fairly called hard, 
so that the law holds tiue. Similarly water must be 
either warm or not-warm, but it does not follow that it 
must be warm or cold. The alternative not-warm evi- 
dently includes all cases in which it is cold besides cases 
where it is of a medium temperature, so that we should 
call it neither warm nor cold. We must thus carefully 
distinguish questions of degree or quantity from those of 
simple logical fact. In cases where a thing or quality 
may exist to a greater or less extent there are many alter- 
natives. Warm water, for, instance may have any tempe- 
rature from 70° perhaps up to 120° Exactly the same 
question occurs in cases uf geometrical reasoning; for 
Euclid in his Elements frequently argues from the self- 
evident truth that any line must be either greater than, 
equal to, or less than any other line. While there are 
only two alternatives to choose from in logic there are 
three in Mathematics; thus one line, compared with 
another, may be — 



/greater greater 

In Logic |not-greater...| ^^^^"^ 



In 

Mathematics. 



Another and even more plausible objection may be 
raised to the third law of thought in this way. Virtue 
being the thing proposed, and tria^igiclar the quaHty, the 
Law of Excluded Middle enables us at once to assert that 
virtue is either triangular or not -triangular. At tirst sight 
it might seem false and absurd to say that an immaterial 
rjotion such as virtue should be either triangular or not, 
because it has nothing in common with those material 
substances occupying space to which the notion of figure 
belongs. But the absurdity would arise, not from any 
falseness in the law, but from misinterpretation of the 
ex/v)ression not-triangular. If in saying that a thing is 



Kiv.] THE LAWS OF THOUGHT, i2t 

^*not triangular'* we are taken to imply that it has some 
figure though not a triangular figure, then of course the 
expression cannot be applied to virtue or anything im- 
material. In strict logic however no such implied mean- 
ing is to be allowed, and not-triangular will include both 
things which have figure other than triangular, as well as 
things which have not the properties of figure at all; and 
it is in the latter meaning that it is applicable to an im- 
material thing. 

These three laws then being universally and neces- 
sarily true to whatever things they are applied, become 
the foundation of reasoning. All acts of reasoning pro- 
ceed from certain judgments, and the act of judgment 
consists in comparing two things or ideas together and 
discovering whether they agree or differ, that is to say 
whether they are identical in any qualities. The laws of 
thought inform us of the very nature of this identity with 
which all thought is concerned. But in the operation 
of discourse or reasoning we need certain additional 
laws, or axioms, or self-evident truths, which may be thus 
stated : 

1. Two terms agreemg with one and the same third 
temt agree with each other. 

2. Two teniis of which 07ie agrees and the other does 
not agree with one and the same third term^ do not agree 
with each other. 

Thes€ self-evident truths are commonly called the 
Canons or Fundamental Principles of Syllogism, and they 
are true whatever may be the kind of agreement in ques- 
tion. The example we formerly used (p. 3) of the a 
greement of the terms ^*Most useful metal" and "cheapest 
metal" with the third common term " Iron," was but 
an instance of the first Canon, and the agreement con- 
sisted in complete identity. In the case of the " Earth," 
the *' Planets," and ^' Bodies revolving in elliptic orbits/' 



122 THE LAWS OF THOUGHT, [less. 

the agreement was less complete, because the Earth is 
only one of many Planets, and the Planets only a small 
portion of all the heavenly bodies, such as Satellites, 
Comets, Meteors, and Double-Stars which revolve in 
such orbits. 

The second of the Canons applies to cases where there 
is disagreement or difference, as in the. following example ; 

Venus is a planet. 

Planets are not self-luminous. 

Therefore Venus is not self-luminous. 

The first of these propositions states a certain agree- 
ment to exist between Venus and planet, just as in the 
previous case of the Earth, but the second proposition 
states a disagreement between Planet and self-luminous 
bodies ; hence we infer a disagreement between Venus 
and self-luminous body. But the reader will carefully 
observe that from two disagreements we ca7t never infer 
anything. If the following were put forth as an argu- 
ment it would be evidently absurd : — 

Sirius is not a planet. 
Planets are not self-luminous. 
Therefore Sirius is not self-luminous. 

Both the premises or propositions given are true, 
and yet the conclusion is false, for all the fixed stars are 
self-luminous, or shine by their own light. We may, in 
fact, state as a third Canon that — 

3. Two terms both disagreeing with 07ie and the 
sa7ne third term may or ?nay not agree with each other. 

Self-evident rules, of an exactly similar nature to these 
three Canons, are the basis of all mathematical reasoning, 
and are usually called axioms. Euclid's first axiom is 
that "Things which are equal to the same thing are equal 
to one another;" and whether we apply it to the length of 
lines, the magnitude of angles, areas, solids, numbers, 



XIV.] THE LAWS OF THOUGHT, 123 

degrees, or anything else which admits of being equal or 
unequal, it holds true. Thus if the lines A and B are each 
equal to C it is evident that each is equal to the other 



A 
B 
C 
D 
E 



Euclid does not give axioms corresponding to the second 
and third Canons, but they are really used in Geometry. 
Thus if ^ is equal to B^ but D is not equal to i?, it follows 
that A is not equal to D^ or things of which one is equal, 
but the other unequal to the same third thing, are unequal 
to each other. Lastly, A and E are two lines both un- 
equal to D and unequal to each other, whereas A and B 
are two lines both unequal to D but equal to each other; 
thus we plainly see that " two things both unequal to the 
same thing may or may not be equal to each other." 

From what precedes it will be apparent that all rea- 
soning requires that there should be one agreement at 
least ; if there be two agreements we may reason to a 
third agreement ; if there be one agreement and one 
difference we may reason to a second difference ; but if 
there be two differences only we cannot reason to any 
conclusion whatever. These self-evident principles will 
in the next Lesson serve to explain some of the rules of 
the Syllogism. 

Logicians however have not confined themselves to 
the use of these Canons, but have often put the same 
truth into a different form in axioms known as the Dicta 
de ornni et millo of Aristotle. This celebrated Latin 
phrase means " Statements concerning all and none," and 
the axiom, or rather pair of axioms, is uf;ually given in 
the following words : 



V. 



124 THE LAWS OF THOUGHT. [less. 

Whatever is predicated of a term distributed whether 
affirmatively or negatively^ may be predicated in like 
manner of everything contained M?ider it. 
Or more briefly : 

What pertains to the higher class pertai?is also to the 
.lower. 

This merely means, in untechnical language, that 
what may be said of all the things of any sort or kind 
may be said of any one or any Dart of those things ; and, 
secondly, what may be denied of all the things in a class 
may be denied of any one or any part of them. What- 
ever may be said of "All planets" may be said of Venus, 
the Earth, Jupiter, or any other planet ; and, as they may 
all be said to revolve in elliptic orbits, it follows that 
this may be asserted of Venus, the Earth, Jupiter, or any 
other planet. Similarly, accoraing to the negative part 
of the Dicta, we may deny that the planets are self- 
luminous, and knowing that Jupiter is a planet may deny 
that Jupiter is self-luminous. A little reflection would 
show that the affirmative Dictum is really the first of the 
Canons in a less complete and general form, and that the 
negative Dictum is similarly tha second Canon. These 
Dicta in fact only apply to such cases of agreement be- 
tween terms as consist in one being the name of a smaller 
class, and another of the larger class containing it. Lo- 
gicians have for the most part strangely overlooked the 
important cases in which one term agrees with another to 
the extent of being identical with it ; but this is a subject 
which we cannot fitly discuss here at any length. It is 
treated in my little work called The Substitution oj 
Similars^. 

Some logicians have held that in addition to the three 
laws which are called the Primary Laws of Thought, 

* Macmillan and Co. 1869, 



XIV.] THE LAWS OF THOUGHT, 125 

there is a fourth called " The Principle or Law of Suffi- 
cient Reason." It was stated by Leibnitz in the following 
words : 

Nothing happens without a reason why it shotdd be 
so rather than otherwise. For instance, if there be a pair 
of scales in every respect exactly alike on each side and 
with exactly equal weights in each scale, it must remain 
motionless and in equilibrium, because there is no reason 
why one side should go down more than the other. It is 
certainly a fundamental assumption in mechanical science 
that if a body is acted upon by two perfectly equal forces 
in different directions it will move equally between them, 
because there is no reason why it should move more to 
one side than the other. Mr Mansel, Sir W. Hamilton 
and others consider however that this law has no place 
in logic, even if it can be held self-evident at all ; and the 
question which appears open to doubt need not be dis- 
cussed here. 

I have so freely used the word axiom in this lesson 
that it is desirable to clear up its meaning as far as pos- 
sible. Philosophers do not perfectly agree about its deri- 
vation or exact meaning, but it certainly comes from the 
verb d|ioa), which is rendered, to thi?tk worthy. It gene- 
rally denotes a self-evident truth of so simple a character 
that it must be assumed to be true, and, as it cannot be 
proved by any simpler proposition, must itself be taken as 
the basis of reasoning. In mathematics it is clearly used 
in this sense. 

See Hamiiton*s Lectures on Logic, Lectures 5 and 6. 



LESSON XV. 

THE RULES OF THE SYLLOGISM. 

Syllogism is the common name for Mediate Inference, 
or inference by a medium or middle term, and is to be 
distinguished from the process of Immediate Inference, or 
inference which is performed without the use of any third 
or middle term. 

We are in the habit of employing a middle term or 
medium whenever we are prevented from comparing tw5 
things together directly, but can compare each of them 
with a certain third thing. We cannot compare the sizes 
of two halls by placing one in the other, but we can 
measure each by a foot rule or other suitable measure, 
which forms a common measure, and enables us to asceiw 
tain with any necessary degree of accuracy their relative 
dimensions. If we have two quantities of cotton goods 
and want to compare them, it is not necessary to bring 
the whole of one portion to the other, but a sample is cut 
off, which represents exactly the quality of one portion, 
and, according as this sample does or does not agree with 
the other portion, so must the two portions of goods agree 
or differ. 

The use of a middle term in syllogism is closely pa- 
rallel to what it is in the above instances, but not exactly 
the same. Suppose, as an example, that we wish to 
ascertain whether or not " Whales are viviparous,'* and 
that we had not an opportunity of observing the fact 
directly ; we could yet show it to be so if we knew that 
"whales are mammalian animals," and that "all mam- 



XV.] THE RULES OF THE SYLLOGISM, 127 

malian animals are viviparous." It would follow that 
" whales are viviparous ; " and so far as the inference is 
concerned it does not matter what is the meaning we 
attribute to the words viviparous and mammalian. In 
this case *' mammalian animal " is the middle term. 

The name Syllogism means the joining together in 
thought of two propositions, and is derived from the 
Greek words avv^ with, and Xoyo?, thought or reason. It 
is thus exactly the equivalent of the word Computation, 
which means thinking together (Latin con^ together, 
puto, to think), or reckoning. In a syllogism we so unite 
in thought two premises, or propositions put forward, that 
we are enabled to draw from them or infer, by means of 
the middle term they contain, a third proposition called 
the conclusion. Syllogism may thus be defined as the 
act of thought by which from two given propositions we 
proceed to a third proposition, the truth of which neces- 
sarily follows from the truth of these given propositions. 
When the argument is fully expressed in language it is 
usual to call it concretely a syllogism. 

The special rules of the syllogism are founded upon 
the Laws of Thought and the Canons considered in the 
previous Lesson. They serve to inform us exactly under 
what circumstances one proposition can be inferred from 
two other propositions, and are eight in number, as 
follows : — 

1. Every syllogism has three and only three te7^7ns. 
These terms are called the major term, the minor 

term, and the middle term. 

2. Every syllogism contains three, and only three 
propositions. 

These propositions are called the major premise, the 
minor premise, and the conclusion. 

3. The middle term must be distributed once at least^ 
and must not be ambiguous. 



128 THE RULES OF THE SYLLOGISM, [less. 

4. No term m^ist be distributed in the conclusion 
which was not distributed in 07ie of the premises, 

5. From negative premises nothing can be infei'red. 

6. If one premise be 7tegative, the conclusion must 
he negative; and vice versa, to prove a 7iegative con^ 
elusion one of the premises must be negative. 

From the above rules may be deduced two subor- 
dinate rules, which it will nevertheless be convenient to 
state at once. 

7. Fro7n two pa^^ticular premises no conclusion can 
he drawn. 

8. If one premise be particular^ the conclusion must 
be particula?'. 

All these rules are of such extreme importance that it 
will be desirable for the student not only to acquire a 
perfect comprehension of their meaning and truth, but to 
commit them to memory. During the remainder of this 
lesson w^e shall consider their meaning and force. 

As the syllogism consists in comparing two terms by 
means of a middle term, there cannot of course be less 
than three terms, nor can there be more ; for if there 
were four terms, say A, B, C, Z>, and we compared A 
with B and C with Z^, we should either have no common 
medium at all between A and D, or we should require a 
second syllogism, so as first to compare A and C with By 
and then A and D with C, 

The middle term may always be knov/n by the fact 
that it does not occur in the conclusion. The major term 
is always the predicate of the conclusion^ and the minor 
term the subject. These terms are thus called because in 
the universal affirmative proposition (A) the predicate is 
necessarily a wider or greater or major term than the 
subject ; thus in " all men are mortals," the predicate in- 
cludes all other animals as well as men, and is obviously 
a major term or wider term than men. 



XV,] THE RULES OF THE SYLLOGISM, 129 

Again, the syllogism necessarily consists of a premise 
called the major premise, in which the major and middle 
terms are compared together ; of a minor premise which 
similarly compares the minor and middle terms ; and of 
a conclusion, which contains the major and m.inor terms 
only. In a strictly correct syllogism the major premise 
always stands before the minor premise, but in ordinary 
writing and speaking this rule is seldom observed ; and 
that premise which contains the major term still con- 
tinues to be the major premise, whatever may be its 
position. 

The third rule is a very important one, because many 
fallacies arise from its neglect. By the middle term being 
distributed once at least, we mean (see p. 74) that the 
whole of it must be referred to universally in one premise, 
if not both. The two propositions — 

All Frenchmen are Europeans, 
All Russians are Europeans, 

do not distribute the middle term at all, because they 
are both affirmative propositions, which have (p. 75) 
undistributed predicates. It is apparent that French- 
men are one part of Europeans, and Russians another 
part, as shown in Euler's method in Fig. 6, so that 

Fig. 6. 




I30 THE RULES OF THE SYLLOGISM, [less. 

there is no real middle term. Those propositions would 
equally allow of .Russians being or not being Frenchmen ; 
for whether the two interior circles overlap or not they 
are equally within the larger circle of Europeans. Again, 
the two propositions 

All Frenchmen are Europeans, 
All Parisians are Europeans, 

do not enable us to infer that all Parisians are French- 
men. For though we know of course that all Parisians 

Fig. 7. 




are included among Frenchmen, the premises would 
allow of their being placed anywhere within the circle of 
Europeans. We see in this instance that the premises 
and conclusion of an apparent argument may all be true 
and yet the argument may be fallacious. 

The part of the third rule which refers to an ambi- 
guous middle term hardly requires expiration. It has 
been stated (Lesson IV.) that an ambiguous term is one 
which has two different meanings, implymg different con- 
notations, and it is really equivalent to two different terms 
which happen to have the same form of spelling, so that 
they are readily mistaken for each other. Thus if we 
were to argue that because " all metals are elements and 



XV.] THE RULES OF THE SYLLOGISM. 131 

brass is metal, therefore it is an element," we should be 
committing a fallacy by using the middle term inetal in 
two different senses, in one of which it means the pure 
simple substances known to chemists as metals, and in 
the other a mixture of metals commonly called metal in 
the arts, but known to chemists by the name alloy. In 
many examples which may be found in logical books the 
ambiguity of the middle term is exceedingly obvious, but 
the reader should always be prepared to meet with cases 
where exceedingly subtle and difficult cases of ambiguity 
occur. Thus it might be argued that " what is right 
should be enforced by law, and that charity is right and 
should therefore be enforced by the law." Here it is 
evident that right is applied in one case to what the 
conscience approves, and in another case to what public 
opinion holds to be necessary for the good of society. 

The fourth rule forbids us to distribute a term in the 
conclusion unless it was distributed in the premises. As 
the sole object of the syllogism is to prove the conclusion 
by the premises, it is obvious that we must not make a 
statement concerning anything unless that thing was 
mentioned in the premises, in a way warranting the state- 
ment. Thus if we were to argue that " because many 
nations are capable of self-government and that nations 
capable of self-government should not receive laws from a 
despotic government, therefore no nation should receive 
laws from a despotic government," we should be clearly 
exceeding the contents of our premises. The minor term, 
many nations^ was particular in the minor premise, and 
must not be made universal in the conclusion. The pre- 
mises d<> not warrant a statement concerning anything but 
the many nations capable of self-government. The above 
argument would therefore be fallacious and would be 
technically called an illicit process of the minor term, 
meaning that we have improperly treated the minor term. 

9—2 



J32 THE RULES OF THE SYLLOGISM, [less. 

Such a breach of the fourth rule as is described above 
is exceedingly easy to detect, and is therefore very seldom 
committed. 

But an illicit process or improper treatment of the 
major term is more common because it is not so trans- 
parently false. If we argued indeed that "because all 
Anglo-Saxons love liberty, and Frenchmen are not Anglo- 
Saxons, therefore they do not love liberty," the fallacy 
would be pretty apparent ; but without a knowledge of 
logic it would not be easy to give a clear explanation of 
the fallacy. It is apparent that the major term loving 
liberty^ is undistributed in the major premise, so that 
Anglo-Saxons must be assumed to be only a part of those 
who love liberty. Hence the exclusion of Frenchmen 
from the class Anglo-Saxons does not necessarily exclude 
them from the class who love liberty (see Fig. 8). The 

Fig. 8. 



Loving Liberty 
Saxons / \^ ^y ^ 



conclusion of the false argument being negative distri- 
butes its predicate, the major term, and as this is un- 
distributed in the major premise we have an Illicit major 
as we may briefly call this fallacy. The following is an 
obscurer example of the same fallacy; — "Few students 



XV.] THE RULES OF THE SYLLOGISM, 133 

are capable of excelling in many brandies of knowledge, 
and such as can so excel are deserving of high commenr 
dation ;" hence " few students are deserving of high com- 
mendation." The little word *' few " has here the double 
meaning before explained (p. 67), and means that '*a 
few are, &c., and the rest are not." The conclusion is 
thus really a negative proposition, and distributes the 
major term "deserving of high commendation." But 
this major term is cle?rly undistributed in the major 
premise, which merely asserts that those who can excel 
in many branches of knowledge are deserving, but says 
or implies nothing about other students. 

The fifth rule is evidently founded on the principle 
noticed in the last lesson, that inference can only proceed 
where there is agreemenv, and that two differences or 
disagreements allow of no reasoning. Two terms, as the 
third Canon states, may both differ from a common term 
and yet may or may not differ from each other. Thus if 

Fig. 9. 




{ Colonists I 



we were to argue that Ainericans are not Europeans, and 
Virginians are not Europeans, we see that both terms 
disagree with the middle term Europeans, and yet they 



134 THE RULES OF THE SYLLOGISM, [LESS. 

agree between themselves. In other cases the two nega- 
tive premises may be plainly true while it \Niil be quite 
uncertain whether the major and minor terms agree or 
not. Thus it is true, for instance, that "Colonists are 
not Europeans, and Americans are not Europeans," but 
this gives us no right to infer that Colonists are or 
are not Americans. The two negative premises are re- 
presented in fig. 9, by excluding the circles of Colonists 
and Americans fixm that of Europeans ; but this exclusion 
may still be effected whether Colonists and Americans 
coincide partially, or wholly, or not at all. A breach of 
this rule of the syllogism may be conveniently called the 
fallacy of negative premises. It must not however be 
supposed that the mere occurrence of a negative particle 
{not or no) in a proposition renders it negative in the 
manner contemplated by this rule. Thus the argument 

" What is not compound is an element. 
Gold is not compound ; 
Therefore Gold is an element." 

contains negatives in both premises, but is nevertheless 
valid, because the negative in both cases affects the middle 
term, which is really the negative term 7iot-compoimd, 

The truth of the sixth rule depends upon that of the 
axiom, that if two terms agree with a common third term 
they agree with each other, whence, remembering that a 
negative proposition asserts disagreement, it is evident 
that a negative conclusion could not be drawn from really 
affirmative premises. The corresponding negative axiom 
prevents our drawing an affirmative conclusion if either 
premise should be really negative. Only practice how- 
ever will enable the student to apply this and the 
preceding rules of the syllogism with certainty, since 
fallacy may be hidden and disguised by various forms of 
expression. Numerous examples are given at the end of 



XV.] THE RULES OF THE SYLLOGISM. 135 

the book by which the student may acquire facihty in 
the analysis of arguments. 

The remaining rules of the syllogism, the 7th and 8th, 
are by no means of a self-evident character and are in 
fact corollaries of the first six rules, that is consequences 
which follow from them. We shall therefore have tc 
shew that they are true consequences in a future Lesson. 
We may call a breach of the 7th rule 2i fallacy of parti- 
cular premises^ and that of the 8th rule the fallacy of a 
U7iiversal conclusion from a parti C2ilar premise^ but these 
fallacies may really be resolved into those of Illicit 
Process, or Undistributed Middle. 

For many details concerning the Aristotelian and 
Scholastic Views of the Syllogism, and of Formal 
Logic generally, see the copious critical notes to 
Mansel's edition of Aldrich's Artis Logiccs Rudi- 
me7tta. 2nd Ed. Oxford. 1852. 



LESSON XVI. 

THE MOODS AND FIGURES OF THE 

SYLLOGISM. 

We are now in full possession of those pringiples of rea- 
soning, and the rules founded upon them, by which a 
true syllogism may be known from one which only seems 
to be a true one, and our task in the present Lesson is to 
ascertain the various shapes or fashions in which a 
process of mediate inference or syllogism may be met 
with. We know that every syllogistic argument must 
contain three propositions and three distinct terms each 
occurring twice in those propositions. Each proposition 



136 THE MOODS AND FIGURES [LESa 

of the syllogism may, so far ars we yet know, be either 
affirmative or negative, universal or particular, so that it 
is not difficult to calculate the utmost possible varieties of 
modes in which a syllogism might conceivably be con- 
structed. Any one of the four propositions A, E, I, or may 
in short be taken as a major premise, and joined with any 
one of the same form as a minor premise, and any one of 
the four again may be added as conclusion. We should 
thus obtain a series of the combinations or modes of 
joining the letters A, E, I, 0, a few of which are here writ 
ten out : 



AAA 


hV.K 


AIA 


AOA 


EAA 


EEA 


AAE 


AEE 


AIE 


AOE 


EAE 


EEE 


AAI 


AEI 


All 


AOI 


EAI 


EEI 


AAO 


AEO 


AIO 


AOO 


EAO 


&c. 



It is obvious that there will be altogether 4x4x4 or 64 
such combinations, of which 23 only are given above. 
The student can easily write out the remainder by carry- 
ing on the same systematic changes of the letters. Thus 
beginning with AAA we change the right-hand letter suc- 
cessively into E, I, and 0, and then do the same beginning 
with AEA instead ; after the middle letter has been carried 
through all its changes we begin to change the left-hand 
letter. With each change of this we have to repeat all 
the sixteen changes of the other letters, so that there will 
obviously be altogether 64 different conceivable modes 
of arranging propositions into syllogisms. 

We call each of these triplets of propositions a mood or 
form of the syllogism (Latin modus ^ shape), and we have 
to consider how many of such forms can really be used in 
valid arguments, as distinguished from those which break 
one or more of the rules of the syllogism. Thus the mood 
AEA would break the 6th rule, that if one premise be 
negative the conclusion must be so too; AIE breaks the 



XVI.J OF THE SYLLOGISM. 137 

converse part of the same rule, that a negative conclusion 
can only be proved by a negative premise; while EEA, 
EEE &c., break the 5th rule, which prohibits our reasoning 
at all from two negative premises. Examples of any of 
these moods can easily be invented, and their falsity would 
be very apparent ; thus for AEA we might take 

All Austrian s are Europeans, 
No Australians are Europeans ; 

Therefore, all Australians are Austrians. 

Many of the 64 conceivable moods are excluded by the 
7th and 8th rules of the syllogism. Thus AIA and EIB 
break the rule, that if one premise be particular the con- 
clusion must be so also, while IIA, 100, 010 and many 
others, break the rule against two particular premises. 
Some combinations of propositions may break more than 
one rule ; thus 000 has both negative premises and parti- 
cular premises, and OOA also violates as well the 6th 
rule. It is an admirable exercise in the use of the syl- 
logistic rules to write out all the 64 combinations and 
then strike out such as break any ru'le ; the task if pur- 
sued systematically will not be so long or tedious as 
might seem likely. It will be found that there are only 
twelve moods which escape exclusion, and may so far be 
considered good forms of reasoning, and these are 

AAA EAE lAI OAO 

AAI EAO (lEO) 

AEE EIO 

AEO 

All 

AOO 

Of these however lEO will have shortly to be rejected, 
because it will be found really to break the 4th rule, and 
involves Illicit process of the major term. There are, 



138 THE MOODS AND FIGURES [less. 

then, only eleven moods of the syllogism which are really 
valid; and we may thus account for the whole of the 
sixty-four moods. 

Number 
Excluded by of moods. 

Negative premises, Rule 5 16 

Particular premises „ 7 ...12 

One negative premise „ 6 12 

One premise particular „ 8 8 

Negative conclusion „ 6 4 

Illicit major „ 4 I 

Total excluded ^'>^ 

Valid moods 11 

Total 64 

We have by no means exhausted as yet all the 
possible varieties of the syllogism, for we have only de- 
termined the character, affirmative or negative, general 
or particular of the propositions, but have not decided 
the ways in which the terms may be disposed in them. 
The major term must be the predicate of the conclusion, 
but it may either be subject or predicate of the major 
premise, and similarly the minor term or subject of the 
conclusion, may be either the subject or predicate of the 
minor premise. There thus arise four different ways, or 
as they are called Figures, in which the terms can be 
disposed. These four figures of the syllogism are shewn 
in the following scheme, taking 

X to denote the major term 

Y middle „ 

Z minor „ 

1st Fig. 2nd Fig. 3rd Fig. 4th Fig. 

Major Premise YX XY YX XY 

Minor „ ZY ZY YZ YZ 

Conclusion ZX ZX ZX ZX 



XVI.] OF THE SYLLOGISM, 139 

These figures must be carefully committed to memory, 
which will best be done by noting the position of the 
middle term. This term stands first as subject of the 
major premise in the ist Figure, second as predicate in 
both premises of the 2nd Figure,yzrj"/ again as subject of 
both premises in the 3rd Figure, and in an intermediate 
position in the 4th Figure. In the conclusion, of course, 
the major and minor terms have one fixed position, and 
when the middle term is once correctly placed in any 
figure we easily complete the syllogism. 

The reader will hardly be pleased to hear that each of 
the eleven valid moods will have to be examined in each 
of the four figures separately, so that there are 44 cases 
still possible, from which the valid syllogisms have to be 
selected. Thuj the mood AEE in the first figure would be 
as follows : 

All F's are X% 

No Z's are F's; %. 

Therefore No Z's are ^'s. 

This would break the 4th rule and be an Illicit Major, 
because X is distributed in the conclusion, which is a 
negative proposition, and not in the major premise. In 
the second figure it would be valid: 

All X's are F's, 
No Z's are F's; 
Therefore No Z's are X^s, 

In the third figure it becomes 

All F's are X% 
No F's are Z's, 
No Z's are ^'s, 

and again breaks the 4th rule, as regards the major term. 
Lastly in the 4th figure it is valid, as the reader may 
easily satisfy himself. 



I40 THE MOODS AND FIGURES [less. 

When all the valid moods are selected out of the 44 
possible ones, there are found to be altogether 24, which 
are as follows: 

Valid Moods of the Syllogism. 

First Second Third Fourth 

Figure. Figure. Figure. Figure. 

AAA EAE AAI AAI 

EAE AEE lAI AEE 

All EIO All lAI 

£10 AOO EAO EAO 

OAO EIO 

[AAI] [EAO] EIO 

[EAO] [AEO] [AEOJ 

Five of the above moods are set apart and enclosed in 
brackets, because though valid they are of little or no use. 
They are said to have a weakened conclusion, because the 
conclusion is particular when a general one might have 
been drawn. Thus AAI, in the first figure is represented 
by the example : 

All material substances gravitate, 
All metals are material substances ; 
Therefore some metals gravitate. 

It is apparent that the conclusion only states a part of 
the truth, and that in reality ad metals gravitate. It is 
not actually an erroneous conclusion, because it must 
be carefully remembered (p. "j^^ that the affirming of a 
subaltern or particular proposition does not deny the 
corresponding general proposition. It is quite true that 
some metals gravitate, and it mast be true because all of 
them do so. But when we can as readily prove that all 
do gravitate it is desirable to adopt this conclusion. 

If we agree v/ith most logicians to overlook the ex- 
istence of the five syllogisms with weakened conclusions, 



XVI.] OF THE SYLLOGISM. 141 

there will remain nineteen which are at once valid and 
useful. In the next lesson certain ancient mnemonic 
lines will be furnished by which alone it would be possible 
for most persons to carry in the memory these 19 combi- 
nations ; but the reader will in the mean time be able to 
gather from the statement of the moods in p. 140 the 
truth of the following remarks concerning the peculiar 
character of each figure of the syllogism. 

The first figure is the only one which proves the pro- 
position A, or has A for its conclusion. It is the only 
figure, too, which can prove any one of the four proposi- 
tions A, E, I, 0. As regards the premises, it is especially 
important to note that the major premise is always 
universal (A or E), and the minor premise affirmative (A or 
I) : this peculiarity will be further considered in the next 
lesson. 

The second figure only proves negative conclusions 
(E or 0), and the reason is easily apparent. As the middle 
term in this figure is the predicate of both premises it 
would necessarily be undistributed in both premises if 
these were affirmatives, and we should commit the fallacy 
exemplified in p. 137. It follows that one premise must 
be negative and of course one only, so that of the major 
and minor terms one must be included or excluded wholly 
from the middle, and the other at the same time excluded 
or included at least partially. To illustrate this we may 
take X^ Y and Z to represent, as before, the major, mid- 
dle and minor terms of a syllogism, and the four moods of 
this figure are then 

EAE AEE 

no X\ are K's, all ^'s are K's, 

all Z's are F's ; no Z's are K's ; 

/. no Z's are JlT's. .*. no Z's are X'^s, 



142 



THE MOODS AND FIGURES 



[less 



EIO 

no X^s are K's, 
some Z's are K's ; 
/. some Z's are not X's. 



AOO 

all X's are Y's, 

some Z^s are not K's ; 

.*. some Z's are not ^*s<, 



The nature of the moods of the second figure is clear!}' 



shewn in the following figures : 



Fig. lo. 
(Cesare.) 



Fig. II. 
(Camestres.) 





Fig. 12. 
(Festino.) 




It will also be obsen'ed that in the second figure the 
minor premise may be any of the four A, E, I. 0. 

The third figure only proves particulars (I or . and 
it always has an aftirmative minor premise (A or I). It 
also contains the greatest number of moods, since in nc 
case is the conclusion a weakened one. 



XVI.] OF THE SYLLOGISM, 143 

The fourth figure is usually considered unnatural and 
comparatively useless, because the same arguments can 
be more clearly arranged in the form of the first figure, 
which in some respects it resembles. Thus it proves all 
the propositions except A, namely, E, I, 0, and its first 
mood AAI, is in reality a weakened form of AAA in the 
first figure. Many logicians, including in recent times 
Sir W. Hamilton, have rejected the use of this figure 
altogether. 

It is evident that the several figures of the syllogism 
possess different characters, and logicians have thought 
that each figure was best suited for certain special pur- 
poses. A German logician, Lambert, stated these pur- 
poses concisely as follows : — "The first figure is suited to 
the discovery or proof of the properties of a thing ; the 
second to the discovery or proof of the distinctions be- 
tween things ; the third to the discovery or proof of in- 
stances and exceptions ; the fourth to the discovery, or 
exclusion, of the different species of genus." 

It may be added that the moods Cesare and Cames- 
tres are often used in disproving a statement, because 
they give a universal negative conclusion, founded upon 
the exclusion of one class from another. Thus if any 
one were still to assert that light consists of material 
particles it might be met by the following syllogism : 

" Material particles communicate impetus to 

whatever they strike, \ 

Light does not communicate impetus to 
whatever it strikes ; 
Therefore light is not material particles." 

The moods Baroko and Festino are less used, but 
allow of a particular conclusion being established. 

When we wish however to establish objections 01 



144 THE IMPERFECT FIGURES [less. 

exceptions to a general statement, which is indeed the 
natural way of meeting it, we employ the third figure. 
The statement that "all metals are solids" would at 
once be disproved by the exception mercury^ as follows : 

Mercury is not solid. 
Mercury is a metal ; 
Therefore some metal is not solid. 

Were any one to assert that what is incomprehensible 
cannot exist, we meet it at once with the argument that 
Infinity is incomprehensible, but that infinity certainly 
exists, because we cannot otherwise explain the nature of 
a curve line, or of a quantity varying continuously ; there- 
fore something that is incom.prehensible exists. In this 
case even one exception is sufficient entirely to negative 
the proposition, which really means that because a thing 
is incomprehensible it cannot exist. But if one incom- 
prehensible thing does exist, others may also ; and all 
authority is taken from the statement. 

According to the Aristotelian system the third figure 
must also be employed whenever the middle term is a 
singular term, because in Aristotle's view of the subject a 
singular term could not stand as the predicate of a pro- 
position. 



LESSON XVII. 

REDUCTION OF THE IMPERFECT FIGURES 
OF THE SYLLOGISM. 

In order to facilitate the recollection of the nineteen valid 
and useful moods of the syllogism, logicians invented, at 
least six centuries ago, a most curious system of artificial 
words, combined into mnemonic verses, which may be 



XVII.] OF THE SYLLOGISM. 145 

readily committed to memory. This device, however in- 
genious, is of a barbarous and wholly unscientific cha- 
racter ; but a knowledge of its construction and use is stilj 
expected from the student of logic, and the verses are 
therefore given and explained below. 

Barbara^ Celarent^ Darii^ Ferioo^Q^ prioris ; 
Cesare^ Camestres^ Festino, Baroko^ secundee; 
Tertia, Darapti, Disainis^ Datisi, Felapton^ 
Bokardo^ Ferison, habet ; Quarta insuper addit 
Brajnantip, Cameiies^ Di7nans, Fesapo, F^^esison. 

The words printed in ordinary type are real Latin 
words, signifying that four moods whose artificial names 
are Barbara, Celarent, Darii and Ferio, belong to the 
first figure ; that four others belong to the second ; six 
more to the third ; while the fourth figure moreover 
contains five moods. Each artificial name contains 
three vowels, which indicate the propositions forming 
a valid mood ; thus, (7E/ArE;// signifies the mood of the 
first figure, which has E for a major premise, A for the 
minor, and E for the conclusion. The artificial words 
altogether contain exactly the series of combinations of 
vowels shown in p. 140, excepting those in brackets. 

These mnemonic lines also contain indications of the 
mode in which each mood of the second, third and fourth 
figures can be proved by reduction to a corresponding 
mood of the first figure. Aristotle looked upon the first 
figure as a peculiarly evident and cogent form of argu- 
ment, the Dictum de oji^ni et nullo being directly ap- 
plicable to it, and he therefore called it the Perfect Figure. 
The fourth figure was never recognised by him, and it is 
often called the Galenian figure, because the celebrated 
Galen is supposed to have discovered it. The second 
and third figures were known to Aristotle as the Imperfect 
Figures, which it was necessary to reduce to the first 



146 THE IMPERFECT FIGURES [les& 

figure by certain conversions and transpositions of the 
premises, for which directions are to be found in the 
artificial words. These directions are as follows : — 

s indicates that the proposition denoted by the pre- 
ceding vowel is to be converted simply, 

p indicates that the proposition is to be converted per 
accidens, or by limitation. 

m indicates that the premises of the syllogism are to 
be transposed, the major being made the minor of a new 
syllogism, and the old minor the new major. The ?n is 
derived from the Latin mtitare, to change. 

B, C, £>, F, the initial consonants of the names, in- 
dicate the moods of the first figure, which are produced 
by reduction; thus Cesare, Camestres and Camenes are 
reducible to "Celarent, Darapti, &c., to Darii, Fresison to 
Ferio and so on. 

^ denotes that the mood must be reduced or proved 
by a distinct process called Indirect reduction, or reductio 
ad impossibile^ which will shortly be considered. 

Let us now take some syllogism, say in Camestres^ and 
follow the directions ^for reduction. Let the example be 

All stars are self-luminous (i) 

All planets are not self-luminous (2) 

Therefore no planets are stars (3) 

Th? first s in Camestres shows that we are to convert 
simply the iiiinor premise. The m instructs us to change 
the order of the premises, and the final s to convert the 
conclusion simply. When all these changes are made 
we obtain 

No self-luminous bodies are planets Converse of (2) 

All stars are self-luminous (i) 

Therefore no stars are planets Converse of (3) 

This, it will be found, is a syllogism in Celarent, as 
might be known from the initial C in Camestres. 



XVII.] OF THE SYLLOGISM, 147 

As another example let us take Fesapo, for instance : 

No fixed stars are planets, 
All planets are round bodres ; 

Therefore some round bodies are not fixed stars. 

According to the directions in the name, we are ta 

convert simply the major premise, and by limitation the 

minor premise. We have then the following syllogism in 

Ferio : 

No planets are fixed stars, 

Some round bodies are planets ; 
Therefore some round bodies are not fixed stars. 

The reader will easily apply the same process of con- 
version or transposition to the other moods, according to 
the directions contained in their names, and the only 
moods it will be necessary to examine especially are 
Bramantip, Baroko and Bokardo. As an example of 
Bramantip we may take : 

All metals are material substances. 

All material substances are gravitating bodies ; 

Therefore some gravitating bodies are metals. 

The name contains the letter ;;/, which instructs us to 
transpose the premises, and the letter /, which denotes 
conversion by limitation ; effecting these changes we 
have : 

All material substances are gravitating bodies, 
All metals are material substances ; 
Therefore some metals are gravitating bodies. 

This is not a syllogism in Barbara, as we might have 
expected, but is the weakened mood AAI of the first 
figure. It is evident that the premises yield the conclusion 
*'all metals are gravitating bodies,^' and we must take the 
letter p to indicate in this mood that the conclusion is 
vveaker than it might be. In truth the fourth figure is so 

io — 2 



148 THE IMPERFECT FIGURES [LESS. 

imperfect and unnatural in form, containing nothing but 
ill-arranged syllogisms, which would have been better 
stated in the first figure, that Aristotle, the founder of 
logical science, never allowed the existence of the figure 
at all. It is to be regretted that so needless an addition 
was made to the somewhat complicated forms of the 
syllogism. 

Indirect reduction. The moods Baroko and Bokardo 
give a good deal of trouble, because they cannot be re- 
duced directly to the first figure. To show the mode of 
treating these moods we will take X^ V, Z to represent the 
major, middle and minor terms of the syllogism, and 
Baroko may then be stated as follows : 

All X's are F's, 
Some Z^s> are not Y's ; 
Therefore Some Z''s are not ^'s. 

Now if we convert the major premise by Contrapo- 
sition (p. 83) we have "all not-F's are not-X's,'' and, 
making this the major premise of the syllogism, we have 

All not- F's are not X's, 
Some Z'^s are not- K's ; 
Therefore Some Z's are not X^s. 

Although both the above premises appear to be nega- 
tive, this is really a valid syllogism in Ferio, because 
two of the negative particles merely affect the middle 
term (see p. 134), and we have therefore effected the re- 
duction of the syllogism. 

Bokardo, when similarly stated, is as follows : 

Some F's are not ^'s, 
All F's are Z's ; 
Therefore Some Z's are not X's. 



xviL] OF THE SYLLOGISM, 149 

To reduce this, convert the major premise by nega- 
tion, and then transpose the premises. We have: 
All K's are Z's, 
Some not-^'s are K's; 
Therefore Some not-X's are Z^s. 

This conclusion is the converse by negation of the 
former conclusion, the truth of which is thus proved by 
reduction to a syllogism in Darii. 

Both these moods, Baroko and Bokardo, may however 
be proved by a peculiar process of indirect reduction, 
closely analogous to the indirect proofs often employed by 
Euclid* in Geometry. This process consists in supposing 
the conclusion of the syllogism to be false, and its con- 
tradictory therefore true, when a new syllogism can easily 
be constructed which leads to a conclusion contradictory 
of one of the original premises. Now it is absurd in logic 
to call in question the truth of our own premises, for the 
very purpose of argument or syllogism is to deduce a con- 
clusion which will be true when the pre77tises are true. 
The syllogism enables us to restate in a n=iw form the in- 
formation which is contained in the premises, just as a 
machine may deliver to us in a new form the material 
which is put into it. The machine, or rather the maker 
of the machine, is not responsible for the quality of the 
materials furaished to it, and similarly the logician is not 
responsible in the least for the truth of his premises, but 
only for their correct treatment. He must treat them, if 
he treat them at all, as true ; and therefore a conclusion 
which requires the falsity of one of our premises is alto- 
gether absurd. 

To apply this method we may take Baroko, as be- 
fore : 

AllJl^'sare F's (i) 

Some Z's are not K's (2) 

Therefore Some Z's are not X's (3) 



ISO THE IMPERFECT FIGURES [less. 

If this conclusion be not true then its contradictory, 
* all Z^s are ^s ' must of necessity be regarded as true 
(pp. "j^i — 79). Making this the minor premise of a new 
syllogism with the original major premise we have : 

All X's are F's (i) 

All Z\s are X^s contradictory of (3) 

Hence All Z's are K's. 

Now this conclusion in A, is the contradictory of our old 
minor premise in 0, and we must either admit one of our 
own premises to be false or allow that our original con- 
clusion is true. The latter is of course the alternative 
we choose. 

We treat Bokardo in a very similar manner ; 

Some F's are not ^s (i) 

All F's are Z^s (2) 

Therefore Some Z's are not ^'s (3) 

If this conclusion be not true then *all Z^s are X's' must 
be true. Now we can make the syllogism : 

All Z's are X'^s Contradictory of (3) 

AllF'sareZ^s (2) 

Hence All Vs are Xs, 

This conclusion is the contradictory of (i), the original 
major premise, and as this cannot be allowed, we must 
either suppose (2) the original minor premise to be false, 
which is equally impossible, or allow that our original 
conclusion is true. 

It will be observed that in both these cases of Indirect 
Reduction or Proof we use a syllogism in Barbara, which 
fact is indicated by the initial letters of Baroko and Bo- 
kardo. The same process of Indirect proof may be 
applied to any of the other moods, but it is not usual to 
do so, as the simpler process of direct or as it is often 
called ostensive reduction is sufficient. 



mmi^mm 



XVII.] OF THE SYLLOGISM. 151 

It will be remembered that when in Lesson xv. (p. 135) 
we considered the rules of the syllogism, there were tvv^o 
supplementary rules, the 7th and 8th, concerning particu- 
lar premises, which were by no means of a self-evident 
character, and which require to be proved by the six more 
fundamental rules. We have now sufficiently advanced 
to consider this proof with advantage. The 7th rule 
forbids us to draw any conclusion from two particular pre- 
mises; now such premises must be either II, 10, 01, or 00. 
Of these n contain no distributed term at all, so that the 
3rd rule, which requires the middle term to be distributed, 
must be broken. The premises 00 evidently break the 
5th rule, against negative premises. The conclusion of 
the pair 10 must be negative by the 6th rule, because one 
premise is negative; the major term therefore will be 
distributed, but ajs the major premise is a particular 
affirmative it cannot be distributed without committing 
the fallacy of illicit process of the major, against rule 4. 
Lastly the premises 01 contain only one distributed term, 
the predicate of the major premise. But as the conclusion 
must be negative by rule 6th, the major term must be 
distributed : we ougnt to have then in the premises two 
distributed terms, one for the middle term, the other for 
the major term; but as the premises contain only a single 
distributed term, we must commit the fallacy either of 
undistributed middle or of illicit process of the major 
term, if we attempt to draw any conclusion at all. We 
thus see that in no possible case can a pair of particular 
premises give a valid conclusion. 

The 8th rule of the syllogism instructs us that if one 
premise of a syllogism be particular the conclusion must 
also be particular. It can only be shown to be true by 
going over all the possible cases and observing that the 
six principal rules of the syllogism always require the 
conclusion to be particular. Suppose for instance the 



152 IRREGULAR AND COMPOUND lLess. 

premises are A and I ; then they contain only one dis- 
tributed term, the subject of A, and this is required for 
the middle term by rule 3. Hence the minor term cannot 
be distributed without breaking rule 4, so that the con- 
clusion must be the proposition I. The premises AO would 
contain two distributed terms, the subject of A and the 
predicate of 0; but if we were to draw from them the 
conclusion E, the major and minor terms would require 
to be distributed, so that the middle term would remain 
undistributed against rule 3. The reader can easily prove 
the other cases such as EI by calculating the number of 
distributed terms in a similar manner: it will always be 
found that there are insufficient terms distributed in the 
premises to allow of a universal conclusion. 



LESSON XVIII. 

IRREGULAR AND COMPOUND SYLLOGISMS. 

It may seem surprising that arguments which are met 
with in books or conversation are seldom or never thrown 
into the form of regular syllogisms. Even if a complete 
syllogism be sometimes met with, it is generally employed 
in mere affectation of logical precision. In former cen- 
turies it was, indeed, the practice for all students at the 
Universities to take part in public disputations, during 
which elaborate syllogistic arguments were put forward 
by one side and confuted by precise syllogisms on the 
other side. This practice has not been very long dis- 
continued at the University of Oxford, and is said to be 
still maintained in some continental Universities ; but 
except in such school disputations it must be allowed that 
perfectly formal syllogisms are seldom employed. 



KViii.] SYLLOGISMS, 153 

In truth, however, it is not syllogistic arguments which 
are wanting; wherever any one of the conjunctions,' 
therefore^ because^ for, since, hefice, inasimtch as, conse-* 
quently occurs, it is certain that an inference is being 
drawn, and this will very probably be done by a tine 
syllogism. It is merely the complete statement of the 
premises and conclusion, which is usually neglected be- 
cause the reader is generally aware of one or other of the 
premises, or he can readily divine what is assumed; and 
it is tedious and even offensive to state at full length what 
the reader is already aware o£ Thus, if I say "atmo- 
spheric air must have weight because it is a material 
substance," I certainly employ a syllogism ; but I think 
it quite needless to state the premise, of which 1 clearly 
assume the truth, that " whatever is a material substance 
has weight." The conclusion of the syllogism is the first 
proposition, viz. "atmospheric air has weight." The 
middle term is " material substance," which does not occur 
in the conclusion; the minor is "atmospheric air," and the 
major, "having weight." Ttie complete syllogism is evi- 
dently : 

All material substances have weight. 
Atmospheric air is a material substance ; 
Therefore atmospheric air has weight. 

This is in the very common and useful mood Barbara. 

A syllogism when incompletely stated is usually called 
an enthjrmeme, and this name is often supposed to be 
derived from two Greek words ((V, in, and 6v\i6<i, mind), 
so as to signify that some knowledge is held by the mind 
a^id is supplied in the form of a tacit, that is a silent or 
understood premise. Most commonly this will be the 
major premise, and then the enthymeme may be said to 
be of the First Order. Less commonly the minor premise 
is unexpressed, and the enthymeme is of the Second 



154 IRREGULAR AND COMPOUND [less. 

Order. Of this nature is the following argument: 
** Comets must be subject to the law of gravitation ; for 
this is true of all bodies which move in elliptic orbits." 
It is so clearly implied that comets move in elliptic orbits, 
that it would be tedious to state this as the minor premise 
in a complete syllogism of the mood Barbara, thus : 

All bodies moving in elliptic orbits are subject to 

the law of gravitation ; 
Comets move in elliptic orbits ; 
Therefore comets are subject to the law of gravitation. 

It may happen occasionally that the conclusion of a 
syllogism is left unexpressed, and the enthymeme may then 
be said to belong to the Third Order. This occurs in the 
case of epigrams or other witty sayings, of which the very 
wit often consists in making an unexpressed truth ap- 
parent. Sir W. Hamilton gives as an instance of this 
kind of enthymeme the celebrated epigram written by 
Porson the English scholar upon a contemporary German 
scholar : 

" The Germans in Greek 

Are sadly to seek ; 

Not five in five score. 

But ninety-five more ; 

All, save only Hermann, 

And Hermann^s a German." 

It is evident that while pretending to make an exception 
of Hermann, the writer ingeniously insinuates that since 
he is a German he has not a correct knowledge of Greek. 
The wonderful speech of Antony over the body of Caesar, 
in Shakspeare's greatest historical play, contains a series 
of syllogistic arguments of which the conclusions are 
suggested only. 

Even a single proposition may have a syllogistic force 
if it clearly suggest to the mind a second premise which 



xviii.] SYLLOGISMS. 155 

thus enables a conclusion to.be drawn. The expression 
of Home Tooke, "Men who have no rights cannot justly 
complain of any wrongs," seems to be a case in point ; for 
there are few people who have not felt wronged at some 
time or other, and they would therefore be likely to argue, 
whether upon true or false premises, as follows : 

Men who have no rights cannot justly complain of 

any wrongs; 
We can justly complain; 
Therefore we are not men who have no rights. 

In other words, we have rights. 

Syllogisms may be variously joined and combined 
together, and it is convenient to have special names for 
the several parts of a complex argument. Thus a syllo- 
gism which proves or furnishes a reason for one of the 
premises of another syllogism is called a Prosyllogism ; 
and a syllogism which contains as a premise the conclu- 
sion of another syllogism is called an Episyllogism. 

Take the example : 

All ^'s are A% 
And all Cs are ^^s ; 
Therefore all Cs are A's. 
But all Us are Cs ; 
Therefore All n% are A's, 

This evidently contains two syllogisms in the mood Bar- 
bara, the first of which is a Prosyllogism with respect to 
the second, while the second is an Episyllogism with 
respect to the first. 

The peculiar name Epicheirema is given to a syllogism 
when either premise is proved or supported by a reason 
implying the existence of an imperfectly expressed pro- 
syllogism ; thus the form, 



156 IRREGULAR AND COMPOUND [less. 

All ^'s are ^'s, for they are /^'s, 
And all C's are ^'s, for they are Qs ; 
Therefore all Cs are ^'s, 

is a double Epicheirema, containing reasons for both 
premises. The reader will readily decompose it into 
three complete syllogisms of the mood Barbara. 

A more interesting form of reasoning is found in the 
chain of syllogisms commonly called the Sorites, from the 
Greek word (TinpoS', meaning heap. It is usually stated in 
this way: 

All ^'s are ^'s, 

All ^'s are C% 

All C's are Us, 

All Us are ^'s ; 
Therefore all ^'s are ^'s. 

The chain can be carried on to any length provided it is 
perfectly consecutive, so that each term except the first 
and last occurs twice, once as subject and once as predi- 
cate. It hardly needs to be pointed out that the sorites 
really contains a series of syllogisms imperfectly ex- 
pressed; thus 

First Syllogism. Second Syllogism. Last Syllogism. 

^'s are Cs, C's are Z^'s, Z)'s are ^'s, 

A's are ^'s ; -^'s are C's; A'^s are Z)'s ; 

.*. ^'s are C's. .*. ^'s are Z>'s. .*. A\ are ^'s. 

Each syllogism furnishes a premise to the succeeding one, 
of which it is therefore the prosyllogism, and any syllo- 
gism may equally be considered the episyllogism of that 
which precedes. 

In the above sorites all the premises were universal 
and affirmative, but a sorites may contain one particular 
premise provided it be the first, and one negative premise 
provided it be the last. The reader may easily assure 
himself by trial, that if any premise except the first were 



XVIII.] SYLLOGISMS. 157 

particular the fallacy of undistributed middle would be 
committed, because one of the middle terms would be the 
predicate of one affirmative premise and the subject of 
another particular premise. If any premise but the last 
were negative there would be a fallacy of illicit process of 
the major term. 

It is not to be supposed that the forms of the syllogism 
hitherto described are all the kinds of reasoning actually 
employed in science or common life. In addition to the 
hypothetical and disjunctive syllogisms and some other 
forms to be described in succeeding lessons, there are 
really many modes of reasoning of which logicians have 
not taken much notice as yet. This was clearly pointed 
out more than two hundred years ago by the writers of 
the Po7't Royal Logic, a work first printed in the year 1662, 
but which has been since reprinted very often and trans- 
lated into a great many languages. The book is named 
from a place near Paris where a small religious com- 
munity lived, of which the authors of the book, namely 
Arnauld and Nicole, and a contributor to it the great 
philosopher and mathematician Pascal, were the most 
celebrated members. The Po7't Royal Logic was to a 
considerable extent the basis of the well-known Witts' 
Logic, but the reader can now be referred to an admirable 
translation of the original work made by Professor Spencer 
Baynes, of St Andrew's. 

Many improvements of Logic may be found in this 
work, such as the doctrine of Extension and Intension 
explained in Lesson v. In the 9th Chapter of the 3rd 
Part moreover it is wisely pointed out that "little pains 
are taken in applying the rules of the syllogism to reason- 
ings of which the propositions are complex, though this 
is often very difficult, and there are many arguments of 
this nature which appear bad, but which are nevertheless 
very good; and besides, the use of such reasonings is 



158 IRREGULAR AND COMPOUND [less. 

much more frequent than that of syllogisms which are 
quite simple." Some examples are given of the complex 
syllogisms here referred to; thus: 

The sun is a thing insensible, 
The Persians worship the sun ; 
Therefore the Persians worship a thing insensible. 

This is an argument which cannot be proved by the rules 
of the syllogism, and yet it is not only evidently true, but 
is an exceedingly common kind of argument. Another 
example is as follows: 

The Divine Law commands us to honour kings; 
Louis XIV. is a king; 

Therefore the Divine Law commands us to honour 
Louis XIV. 

The reader will also find that arguments which are 
really quite valid and syllogistic are expressed in language 
so that they appear to have four distinct terms and thus to 
break one of the rules of the syllogism. Thus if I say 
** Diamonds are combustible, for they are composed of 
carbon and carbon is combustible," there are four terms 
employed, namely, diamonds, combustible, composed of 
carbon, and carbon. But it is easy to alter the construc- 
tion of the propositions so as to get a simple syllogism 
without really altering the sense, and we then have : 

What is composed of carbon is combustible ; 
Diamonds are composed of carbon; 
Therefore diamonds are combustible. 

Examples are given at the end of the book of concise 
arguments, taken from Bacon's Essays and other writings, 
which the student can reduce to the syllogistic forni by 
easy alterations ; but it should be clearly understood that 
tliese changes are of an extra-logical character, and belong 
more properly to the science of language. 



XVIII.] SYLLOGISMS, 159 

I may here explain that the syllogism and the sorites 
can be expressed either in the order of extension or that 
of intension. In regard to the number of individual 
things the noble metals are part of the metals, and the 
metals are part of the elements ; but in regard to in- 
tension, that is to say the qualities implied in the names, 
element is part of metal, and metal is part of noble metal. 
So again in extension the genus of plants Anemone is 
part of the order Ranunculaceae, and this is part ol 
the great class Exogens; but in intension the cha- 
racter of Exogen is part of the character of Ranuncu- 
laceae, and this is part of the character of Anemone. 
Syllogistic reasoning is equally valid and evident in either 
case, and we might represent the two modes in ordinary 
language as follows : 

Extensive Syllogism, 
All Ranunculaccce are Exogens ; 
The Anemone is one of the Ranunculaceae ; 
Therefore the Anemone is an Exogen. 

hiteiisive Syllogism. 

All the qualities of Ranunculaceae are qualities of 
Anemone ; 

All the qualities of Exogen are qualities of Ranun- 
culaceae ; 

Therefore all the qualities of Exogen are qualities of 
Anemone. 

Any sorites can be similarly represented either in ex- 
tension or intension. 

Concerning the Aristotehan doctrine of the Enthy- 
meme, see Mansel's Aldrich, App. Note F, and Hamil- 
ton's Lectures on Logic, Lecture XX. Port Royal Logic, 
translated by T. Spencer Baynes, 5th ed. Edinburgh^ 
1861. 



i6o OF CONDITIONAL [less, 

LESSON XIX. 

OF CONDITIONAL ARGUMENTS. 

It will be remembered that when treating of propositions 
tve divided them into two distinct kinds, Categorical Pro- 
positions, and Conditional Propositions. The former kind 
alone has hitherto been considered, and we must now 
proceed to describe Conditional propositions and the ar- 
guments which may be composed of them. 

Logicians have commonly described Conditional pro- 
positions as composed of two or more Catego) ical pro- 
'i)ositions united by a conjunction. This union may 
happen in two ways, giving rise to two very different 
species of conditionals, which we shall call Hypothetical 
Propositions and Disjunctive Propositions. The way in 
which the several kinds of propositions are related will 
be seen in the following diagram : 

( Categorical. 
Propositions are (Hypothetical. 

^ Conditional] T^. • .. 

/Disjunctive. 

A conditional proposition may be further described 
as one which makes a statement under a certain con- 
dition or qualification restricting its application. In the 
hypothetical form this condition is introduced by the 
conjunction if, or some other word equivalent to it. 

Thus — 

"If iron is impure, it is brittle " 

is a hypothetical proposition consisting of two distinct 
categorical propositions, the first of which, " Iron is im- 
pure,'' is called the Antecedent; the second, " It is brittle," 



XIX.J ARGUMENTS. i6i 

the Consequent. In this case " impurity " is the condition 
or qualification which limits the application of the pre- 
dicate brittle to iron. It was asserted by Home Tooke in 
his celebrated work The Diversions of Purley^ that all 
conjunctions are the remains or corrupted forms of verbs. 
This is certainly true in the case of the hypothetical con- 
junction ; for the w^ord if in old English is written gif or 
gyf and is undoubtedly derived from the verb to give. 
We may actually substitute at present any verb of similar 
meaning, as for instance — grants allow^ suppose. Thus 
we may say — 

" Grant that iron is impure, and it is brittle." 
" Supposing that iron is impure, it is brittle." 

The hypothetical proposition might be employed in 
arguments of various form, but only two of these are of 
sufficient importance to receive special names. The hy- 
pothetical syllogism consists of two premises, called the 
major and minor, as in the case of the ordinary syllo- 
gism. The major premise is hypothetical in form ; the 
minor premise is categorical, and according as it is af- 
firmative or negative the argument is said to be a Construc- 
tive or a Destructive hypothetical syllogism. Thus the form, 

li A is B, C is D\ 
But A is B; 
Therefore C is Z>, 

is a constructive hypothetical syllogism. 

It must be carefully observed that the minor premise 
affirms the antecedent of the major premise, whence the 
argument is said to be of the modus ponetts, or mood 
which posits or affirms. It is probably one of the most 
familiar and common kinds of argument. The form, 

li A is B, C is D; 

But C is not D ; 
Therefore A is not B, 



1 62 OF CONDITIONAL [less, 

represents the corresponding Dsstructive hs^potlietical 
syllogism, also called the modus tollens, or the mood 
which removes the consequent. It must be carefully ob- 
served again that it is the consequent, not the antecedent, 
which is denied. 

The only rule which is requisite for testing the validity 
of such syllogisms embodies what we have observed 
above ; viz. tRat either the antecedent must be affinned^ 
or the consequent denied. If either part of this rule be 
broken, a serious fallacy will be committed. Thus the 
apparent argument, 

\i A \s B, C \s D \ 
But C is Z^ ; 
Therefore A is B, 

is really a fallacy which we may call \k\Q, fallacy of affirm- 
ing the conseqtient^ and its fallacious nature is readily un- 
derstood by reflecting that " A being B " is not stated to 
be the only condition on which C is Z>. It may happen 
that when E is F^ or G is H^ or under a hundred other 
circumstances, C is Z>, so that the mere fact of C being D 
is no sufficient proof that A is B, Thus, if a man's cha- 
racter be avaricious he will refuse to give money for useful 
purposes ; but it does not follow that every person who 
refuses to give money for such purposes is avaricious. 
There may be many proper reasons or motives leading 
him to refuse ; he may have no money, or he may con- 
sider the purpose not a useful one, or he may have more 
useful purposes in view. 

A corresponding fallacy arises from denying the xnt^" 
udenty as in the form — 

If ^ is ^, C is Z? ; 
But A is not B \ 
Therefore C is not D, 



XIX.] ARGUMENTS, 163 

The error may be explained in the same way ; for as 
*M being j5" is not stated to be the only condition of 
C being Z>, we may deny this one condition to be true^ 
but it is possible that the consequent may happen to be 
true for other reasons, of which we know nothing. Thus 
if a man is not avaricious we cannot conclude that he will 
be sure to give money whenever asked. Or take the fol- 
lowing example : 

"If the study of Logic furnished the mind with a multi- 
tude of useful facts like the study of other sciences, it 
would deserve cultivation; but it does not furnish the 
mind with a multitude of useful facts; therefore it does 
not deserve cultivation." 

This is evidently a fallacious argument, because the 
acquiring of a multitude of useful facts is not the only 
ground on which the study of a science can be recom- 
mended. To correct and exercise the powers of judgment 
and reasoning is the object for which Logic deserves to 
be cultivated, and the existence of such other purpose is 
ignored in the above fallacious argument, which evidently 
involves the denial of the antecedent. 

Although it is usual in logical works to describe the 
hypothetical proposition and syllogism as if they were 
different in nature from the categorical proposition and 
syllogism, yet it has long been known that the hypo- 
theticals can be reduced to the categorical form, and 
brought under the ordinary rules of the syllogism. As a 
general rule the hypothetical proposition can be readily 
converted into a universal affirmative proposition (A) of 
exactly the same meaning. Thus our instance, "If iron 
is impure, it is brittle," becomes simply "Impure iron is 
brittle." In making this alteration in a hypothetical syl- 
logism it will be found necessary to supply a new minor 
term ; thus ui the case, 

1 1 — 2 



i64 OF CONDITIONAL [LESa 

If iron is impure it is brittle ; 
But it is impure ; 
Therefore it is brittle, 

we have to substitute for the indefinite pronoun //, the 
iron in question^ and we obtain a correct categorical syl- 
logism in the mood Barbara : 

Impure iron is brittle ; 

The iron in question is impure iron ; 

Therefore the iron in question is brittle. 

Sometimes the reduction requires a more extensive 
change of language. For instance, 

If the barometer is falling, bad weather is coming ; 
But the barometer is falling ; 
Therefore bad weather is coming, 

may be represented in the following form : 
The circumstances of the barometer falling are the cir- 
cumstances of bad weather coming ; 
But these are the circumstances of the barometer fall- 
ing; 
Therefore these are the circumstances of bad weather 
coming. 
As an instance of the Destructive Hypothetical syl- 
logism we may take : 
If Aristotle is right, slavery is a proper form of society; 
But slavery is not a proper form of society; 
Therefore Aristotle is not right. 
This becomes as a categorical : 
The case of Aristotle being right is the case of slavery 

being a proper form of society; 
But this is not the case ; 
I'herefore this is not the case of Aristotle being right. = 

If not reducible by any other form of expression, hypo^ 
theticals can always be reduced by the use of the words 
case of. 



XIX.] ARGUMENTS. 165 

It will now be easily made apparent that the fallacy of 
affirming the consequent is really a breach of the 3fd- 
rule of the syllogism, leading to an undistributed middle 
term. Our example may be as before ; 

If a man is avaricious he will refuse money ; 

But he does refuse money ; 

Therefore he is avaricious. 

This becomes as a categorical syllogism, 

All avaricious men refuse money; 
But this man refuses money ; 
Therefore this man is avaricious. 

This is the mood AAA in the second figure ; and the 
niddle term, refusing money, is undistributed in both 
premises, so that the argument is entirely fallacious. 

Again, the fallacy of denying the antecedent is equiva- 
lent to the illicit process of the major. Our former 
example (p. 163) may thus be represented: 

"A science which furnishes the mind with a multitude 
of useful facts deserves cultivation ; but Logic is not such 
a science ; therefore Logic does not deserve cultivation." 

This apparent syllogism is of the mood AEE in the 
first figure, which breaks the fourth rule of the syllogism, 
because the major term, deserving cultivatiojt^ is dis- 
tributed in the negative conclusion, but not in the affirma- 
tive major premise. 

We now pass to the consideration of the disjunctive 
proposition, which instead of a single predicate has 
several alternatives united by the disjunctive conjunction 
or^ any one of which may be affirmed of the subject. "A 
member of the House of Commons is either a representa- 
tive of a county, or of a borough, or of a University," is an 
mstance of such a proposition, containing three alterna- 
tives ; but there may be any number of alternatives from 
two upwards. 




i66 OF CONDITIONAL [less. 

The disjunctive syllogism consists of a disjunctive 
major premise with a categorical proposition, either af- 
firmative or negative, forming the minor premise. Thus 
arise two moods, of which the affirmative mood is called 
by the Latin words modus ponendo tollens (the mood 
which by affirming denies), and may be thus stated : 

A is either B or C, 

But ^ is ^ ; 

Therefore A is not C 
This form of argument proceeds on the supposition 
that if one alternative of a disjunctive proposition be held 
true, the others cannot also be true. Thus " the time of 
year must be either spring, summer, autumn or winter," 
and if it be spring it cannot be summer, autumn or winter ; 
and so on. But it has been objected by Whately, Man- 
sel. Mill, as well as many earlier logicians, that this does 
not always hold true. Thus if we say that "a good book 
is valued either for the usefulness of its contents or the 
excellence of its style," it does not by any means follow 
because the contents of a book are useful that its style is 
not excellent. We generally choose alternatives which 
are inconsistentjvith each other; but this is not logically 
necessary. 

The other form of disjunctive syllogism, called the 
modus tollendoponens (the mood which by den^ng affirms), 
is always of necessity cogent, and is as follows : 

A is either B ox C^ 
But ^ is not ^ ; 
Therefore A is C 

Thus if we suppose a book to be valued only for the 
usefulness of its contents or the excellence of its style, it 
follows that if a book be valued but not for the former 
reason it must be for the latter ; and vice versa. If the 
time of year be not spring, it must be summer, autumn or 



XIX.] ARGUMENTS, i6r 

winter ; if it be not autumn nor winter, it must be either 
spring or summer; and so on. In short if any alternatives 
be denied, the rest remain to be affirmed as before. It 
will be noticed that the disjunctive syllogism is governed 
by totally different rules from the ordinary categorical 
syllogism, since a negative premise gives an affirmative 
conclusion in the former, and a negative conclusion in 
the latter. "^^^ ^^^ 

There yet remains a form of argument called the 
Dilemma, because it consists in assuming two alternatives, 
usually called the horns of the dilemma, and yet proves 
something in either case (Greek hi- two ; Xijfifxa, assump- 
tion). Mr Mansel defines this argument as " a syllogism, 
having a conditional major premise with more than one 
antecedent, and a disjunctive minor. *' There are at least 
three forms in which it may be stated. The first form is 
nailed the Simple Constructive Dilemma : 

If ^ is ^, C is Z? ; and if ^ is i% C is D ; 
But either A is B, or E is E; 
Therefore C is D, 

Thus "if a science furnishes useful facts, it is worthy of 
being cultivated; and if the study of it exercises the 
reasoning powers, it is worthy of being cultivated ; but 
either a science furnishes useful facts, or its study 
exercises the reasoning powers ; therefore it is worthy of 
being cultivated." 

The second form of dilemma is the Complex Con 
structive Dilemma, which is as follows : 

If ^ is ^, C is Z> ; and if ^ is 7% G is H; 
But either ^ is ^, or ^5" is i^; 
Therefore either C is D, or G is //, 

It is called complex because the conclusion is in the 
disjunctive form. As an instance we may take the argu- 



i68 OF CONDITIONAL [LESS, 

ment, " If a statesman who sees his former opinions to 
be wrong does not alter his course he is guilty of deceit; 
and if he does alter his course he is open to a charge 
of inconsistency ; but either he does not alter his course 
or he does ; therefore he is either guilty of deceit, or he is 
open to a charge of inconsistency." In this case as in 
the greater number of dilemmas the terms A^B^ C, Z>, &;c. 
are not all different. 

The Destructive Dilemma is always complex, becaus*e 
it could otherwise be resolved into two unconnected de- 
structive hypothetical syllogisms. It is in the following 
form ; 

If A is B, C is D; and if E is F, G is H; 
But either C is not £>, or G is not H; 
Therefore either A is not B, or E is not E, . 

For instance, " If this man were wise, he would not 
speak irreverently of Scripture in jest ; and if he were 
good, he would not do so in earnest ; but he does it either 
in jest or earnest ; therefore he is either not wise, or not 
good *.^' 

Dilemmatic arguments are however more often fal- 
lacious than not, because it is seldom possible to find 
instances where two alternatives exhaust all the possible 
cases, unless indeed one of them be the simple negative 
of the other m accordance with the law of excluded mid- 
dle (p. 1 19). Thus if we were to argue that " if a pupil is 
fond of learning he needs no stimulus, and that if he dis- 
likes learning no stimulus will be of any avail, but as he 
is either fond of learning or dislikes it, a stimulus is either 
needless or of no avail," we evidently assume improperly 
the disjunctive minor premise. Fondness and dislike are 
not the only two possible alternatives, for there may be 

♦ Whately. 



XIX.] ARGUMENTS. 169 

some who are neither fond of learning nor dislike it, and 
to these a stimulus in the shape of rewards may be de- 
sirable. Almost anything can be proved if we are allowed 
thus to pick out two of the possible alternatives which are 
in our favour, and aigue from these alone. 

A dilemma can often be retorted by producing as 
cogent a dilemma to the contrary effect. Thus an Athe- 
nian mother, according to Aristotle, addressed her son in 
the following words : " Do not enter into public business ; 
for if you say what is just, men will hate you ; and if you 
say what is unjust, the Gods will hate you." To which 
Aristotle suggests the following retort : " I ought to enter 
into" public affairs ; for if I say what is Just, the Gods will 
love me ; and if I say what is unjust, men will love me." 

ManseFs Aldrich, App. Note I, on the Hypothetical 
Syllogism. 



LESSON XX. 

LOGICAL FALLACIES. 

In order to acquire a satisfactory knowledge of the rules 
of correct thinking, it is essential that we should become 
acquainted with the most common kinds of fallacy ; that 
is to say, the modes in which, by neglecting the rules of 
logic, we often fall into erroneous reasoning. In previous 
lessons we have considered, as it were, how to find the 
right road ; it is our task here to ascertain the turnings at 
which we are most liable to take the wrong road. 

In describing the fallacies I shall follow the order and 
adopt the mode of classification which has been usual 
for the last 2000 years and more, since in fact the great 



I70 LOGICAL FALLACIES, [less. 

teacher Aristotle first explained the fallacies. According 
to this mode of arrangement fallacies are divided into two 
principal groups, containing the logical and the material 
fallacies. 

1. The logical fallacies are those which occur in the 
mere form of the statement ; or as it is said in the old 
Latin expressions, in dictione^ or in voce. It is supposed 
accordingly that fallacies of this kind can be discovered 
without a knowledge of the subject-matter with which the 
argument is concerned. 

2. The material fallacies, on the contrary, arise out- 
side of the mere verbal statement, or as it is said^ extra 
dictionem; they are concerned consequently with the sub- 
ject of the argument, or m re (in the matter), and cannot 
be detected and set right but by those acquainted with 
the subject. 

The first group of logical fallacies may be further di- 
vided into the purely logical and the semi-logical^ and we 
may include in the former class the distinct breaches of 
the syllogistic rules which have already been described. 
Thus we may enumerate as Purely Logical Fallacies : 

1. Fallacy of four terms {Qiiaternio Terminorum) — 
Breach of Rule i ; 

2. Fallacy of undistributed middle — Breach of Rule 3 ; 

3. Fallacy of illicit process, of the major or minor 
term — Breach of Rule 4 ; 

4. Fallacy of negative premises — Breach of Rule 5 ; 
as well as breaches of the 6th rule, to which no distinct 
name has been given. Breaches of the 7th and 8th rules 
may be resolved into the preceding (p. 151), but they 
may also be described as in p. 135. 

The other part of the class of logical fallacies contains 
Semi-logical fallacies, which are six in number, as follows ? 



XX.] LOGICAL FALLACIES, 171 

1. Fallacy of Equivocation. 

2. Fallacy of Amphibology. 

3. Fallacy of Composition. 

4. Fallacy of Division. 

5. Fallacy of Accent. 

6. Fallacy of Figure of Speech. 

These I shall describe and illustrate in succe,ssion. 

Equivocation consists in the same term being used 
in two distinct senses ; any of the three terms of the syl- 
logism may be subject to this fallacy, but it is usually the 
middle term which is used in one sense in one premise 
and in another sense in the other. In this case it is often 
called the fallacy of ambiguous 7niddle^ and when we dis- 
tinguish the two meanings by using other suitable modes 
of expression it becomes apparent that the supposed syl- 
logism contains four terms. The fallacy of equivocation 
may accordingly be considered a disguised fallacy of four 
terms. Thus if a person were to argue that " all criminal 
actions ought to be punished by law ; prosecutions for 
theft are criminal actions; therefore prosecutions for 
theft ought to be punished by law," it is quite apparent 
that the term " criminal action " means totally different 
things in the two premises, and that there is no true 
middle term at all. Often, however, the ambiguity is of 
a subtle and difficult character, so that different opinions 
may be held concerning it. Thus we might argue : 

" He who harms another should be punished. He 
who communicates an infectious disease to another per- 
son harms him. Therefore he who communicates an 
infectious disease to another person should be punished.'* 

This may or may not be held to be a correct argument 
according to the kinds of actions we should consider to 
come under the term kar7n, according as we regard negli- 
gence or malice requisite to constitute harm. Many 



172 LOGICAL FALLACIES. [LEy:> 

difficult legal questions are of this nature, as for in- 
stance : 

Nuisances are punishable by law ; 

To keep a noisy dog is a nuisance ; 

To keep a noisy dog is punishable by law. 

The question here would turn upon the degree of 
nuisance which the law would interfere to prevent. Or 
again : 

Interference with another man's business is illegal; 
Underselling interferes with another man's business; 
Therefore underselling is illegal. 

Here the question turns upon the kmd of interfereiice^ 
and it is obvious that underselling is not the kind of in- 
terference referred to in the major premise. 

The Fallacy of Amphibology consists in an ambiguous 
grammatical structure of a sentence, which produces mis- 
conception. A celebrated instance occurs in the prophecy 
of the spirit in Shakspeare's Henry VI.: "The Duke yet 
lives that Henry shall depose," which leaves it wholly 
doubtful whether the Duke shall depose Henry, or Henry 
the Duke. This prophecy is doubtless an imitation of 
those which the ancient oracle of Delphi is reported to 
have uttered ; and it seems that this fallacy was a great 
resource to the oracles who were not confident in their 
own powers of foresight. The Latin language gives great 
scope to misconstructions, because it does not require 
any fixed order for the words of a sentence, and when 
there are two accusative cases with an infinitive verb, it 
may be difficult to tell except from the context which 
comes in regard to sense before the verb. The double 
meaning which may be given to ^' twice two and three" 
arises from amphibology ; it may be 7 or 10, according 
as we add the 3 after or before multiplying. In the 
careless construction of sentences it is often impossible to 



KX.J LOGICAL FALLACIES, 17: 

tell to what part any adverb or qualifying clause refers. 
Thus if a person says " I accomplished my business and 
returned the day after," it may be that the business was 
accomplished on the day after as well as the return ; but 
it may equally have been finished on the previous day. 
Any ambiguity of this kind may generally be avoided by 
a simple change in the order of the words ; as for instance, 
" I accomplished my business, and, on the day after, 
returned." Amphibology may sometimes arise from con- 
fusing the subjects and predicates in a compound sentence, 
as if in "platinum and iron are very rare and useful 
metals " I were to apply the predicate useful to platinum 
and rare to iron, which is not intended. The word " re- 
spectively" is often used to shew that the reader is not at 
liberty to apply each predicate to each subject. 

The Fallacy of Composition is a special case of equivo- 
cation, arising from the confusion of an universal and a 
collective term. In the premises of a syllogism we may 
affirm something of a class of things distribiitively^ that is, 
of each and any separately, and then we may in the con- 
clusion infer the same of the whole put together. Thus we 
may say that ^' all the angles of a triangle are less than two 
right angles," meaning that aiiy of the angles is less than 
two right angles ; but we must not infer that all the angles 
put together are less than two right angles. We must not 
argue that because every member of a jury is very likely 
to judge erroneously, the jury as a whole are also very 
likely to judge erroneously ; nor that because each of the 
witnesses in a law case is liable to give false or mis- 
taken evidence, no confidence can be reposed in the con- 
current testimony of a number of witnesses. It is by a 
fallacy of Composition that protective duties are still 
sometimes upheld. Because any one or any few trades 
which enjoy protective duties are benefited thereby, it is 
supposed that all trades at once might be benefited simi- 



174 LOGICAL FALLACIES. [less. 

larly; but this is impossible, because the protection of one 
trade by raising prices injures all others. 

The Fallacy of Division is the converse of the pre- 
ceding, and consists in using the middle term col- 
lectively in the major premise but distributively in the 
minor, so that the whole is divided into its parts. Thus 
it might be argued, *'A11 the angles of a triangle are 
(together) equal to two right angles; ABC is an angle of 
a triangle ; therefore ABC is equal to two right angles." 
Or again, " The inhabitants of the town consist of men, 
women and children of all ages ; those who met in the 
Guildhall were inhabitants of the town; therefore they 
consisted of men, women and children of all ages;" or, 
" The judges of the court of appeal cannot misinterpret 
the law; Lord A. B. is a judge of the court of appeal; 
therefore he cannot misinterpret the lav/." 

The Fallacy of Accent consists in any ambiguity 
arising from a misplaced accent or emphasis thrown upon 
some word of a sentence. A ludicrous instance is liable 
to occur in reading chapter xiii. of the First Book of 
Kings, verse 27, where it is said of the prophet "And he 
spake to his sons, saying, Saddle me the ass. And they 
saddled him.^'* The italics indicate that the word hhn 
was supplied by the translators of the authorized version, 
but it may suggest a very different meaning. The Com- 
mandment " Thou shalt not bear false witness against 
thy neighbour " may be made by a slight emphasis of the 
voice on the last word to imply that we are at liberty to 
bear false witness against other persons. Mr De Morgan 
who remarks this also points out that the erroneous 
quoting of an author, by unfairly separating a word from 
its context or italicising words which were not intend,ed 
to be italicised, gives rise to cases of this fallacy. 

It is curious to observe how many and various may be 
the meanings attributable to the same sentence according 



XX.1 LOGICAL FALLACIES. 175 

as emphasis is thrown upon one word or another. Thu< 
the sentence " The study of Logic is not supposed to 
communicate a knowledge of many useful facts," may be 
made to imply that the study of Logic does communicate 
such a knowledge although it is not supposed to ; or that 
it communicates a knowledge of a few useful facts ; or 
that it communicates a knowledge of many useless facts. 
This ambiguity may be explained by considering that if 
you deny a thing to have the group of qualities A^B,C, D, 
the truth of your statement will be satisfied by any one 
quality being absent, and an accented pronunciation will 
often be used to indicate that which the speaker believes 
to be absent. If you deny that a particular fruit is ripe 
and sweet and well-havoured, it may be unripe and sweet 
and well-flavoured ; or ripe and sour and well-flavour- 
ed; or ripe and sweet and ill-flavoured; or any two or 
even all three qualities may be absent. But if you deny 
it to be ripe and sweet and well-flavoured^ the denial 
would be understood to refer to the last quality. Jeremy 
Bentham was so much afraid of being misled by this 
fallacy of accent that he employed a person to read to 
him, as I have heard, who had a peculiarly monotonous 
manner of reading. 

The Fallacy of the Figure of Speech is the sixth and 
last of the semi-logical fallacies, and is of a very trifling 
character. It appears to consist in any grammatical 
mistake or confusion between one part of speech and an- 
other. Aristotle gravely gives the following instance : 
" Whatever a man walks he tramples on ; a man walks 
the whole day; therefore he tramples on the day." Here 
an adverbial phrase is converted into a noun object. 



LESSON XXI. 

MATERIAL FALLACIES. 

The Material fallacies are next to be considered; and theit 
importance is very great, although it is not easy to 
illustrate them by brief examples. There are altogether 
seven kinds of such fallacies enumerated by Aristotle and 
adopted by subsequent logicians, as follows : 

1. The Fallacy of Accident. 

2. The Converse Fallacy of Accident. 

3. The Irrelevant Conclusion. 

4. The Petitio Principii. 

5. The Fallacy of the Consequent or Non sequitur. 

6. The False Cause. 

7. The Fallacy of Many Questions. 

Of these the two first are conveniently described to- 
gether. The fallacy of a.ccident consists in arguing erro- 
neously from a general rule to a special case, where a 
certain accidental circumstance renders the rule inappli- 
cable. The converse fallacy consists in arguing from a 
special case to a general one. This latter fallacy is usu- 
ally described by the Latin phrase a dicto^ secundum quid 
ad dictuin simpliciter^ meaning " from a statement under 
a condition to a statement simply or without that con- 
dition." Mr De Morgan has remarked in his very inte- 
resting Chapter on Fallacies"^ that we ought to add a 
third fallacy, which would consist in arguing from ons 
special case to another special case, 

* Formal Logic, Chapter XIII. 



LESS. XXL] MATERIAL FALLACIES, 17} 

I will try by a few examples to illustrate these kinds of 
fallacy, but much difficulty is often encountered in saying 
to which of the three any particular example is best re- 
ferred. A most ancient example repeated in almost every 
logical hand-book is as follows : " What you bought yes 
terday you eat to-day ; you bought raw meat yesterday ; 
therefore you eat raw meat to-day." The assertion in the 
conclusion is made of meat with the accidental quality of 
rawness added, where the first premise evidently speaks of 
the substance of the meat without regard to its accidental 
condition. This then is a case of the direct fallacy. 
If it is argued again that because wine acts as a poison 
when used in excess it is always a poison, we fall into the 
converse fallacy. 

It would be a case of the direct fallacy of accident 
to infer that a magistrate is justified in using his power 
to forward his own religious views, because every man 
has a right to inculcate his own opinions. Evidently 
a magistrate as a man has the rights of other men, but 
in his capacity of a magistrate he is distinguished from 
other men, and he must not infer of his special powers 
in this respect what is only true of his rights as a 
man. For another instance take the following : "He who 
thrusts a knife into another person should be punished ; 
a surgeon in operating does so ; therefore he should be 
punished." Though the fallacy of this is absurdly 
manifest, it is not so manifest how we are to classify the 
error. We may for instance say that as a general rule 
whoever stabs or cuts another is to be punished unless it 
can be shewn to have been done under exceptional cir- 
cumstances, as by a duly qualified surgeon acting for the 
good of the person. In this case the example belongs to 
the direct fallacy of accident. In another view we might 
interpret the first premise to mean the special case o\ 
thrusting a knife jnaliciouslyj to argue from that to the 

12 



178 MATERIAL FALLACIES, [less. 

case of a surgeon would be to infer from one special case 
to another special case. 

It is undoubtedly true that to give to beggars promotes 
mendicancy and causes evil ; but \i we interpret this to 
mean that assistance is never to be given to those who 
solicit it, we fall into the converse fallacy of accident, 
inferring of all who solicit alms what is only true of those 
who solicit alms as a profession. Similarly it is a very 
good rule to avoid lawsuits and quarrels, but only as a 
general rule, since there frequently arise circumstances 
in which resort to the law is a plain duty. Alm.ost ali 
the difficulties which we meet in matters of law ana 
moral duty arise from the impossibility of always ascer- 
taining exactly to what cases a legal or moral rule does 
or does not extend ; hence the interminable differences 
of opinion, even among the judges of the land. 

The Third Material Fallacy is that of the IiTelevant 
Conclusion, technically called the Ignoratio Elenchi^ or 
literally Ignorance of the Refutation. It consists in 
arguing to the wrong point, or proving one thing in such 
a manner that it is supposed to be something else that is 
proved. Here again it would be difficult to adduce con- 
cise examples, because the fallacy usually occurs in the 
course of long harangues, where the multitude of words 
and figures leaves room for confusion of thought and 
forgetfulness. This fallacy is in fact the great resource of 
those who have to support a v;-eak case. It is not un- 
known in the legal profession, and an attorney for the 
defendant in a lawsuit is said to have handed to 
the barrister his brief marked, "No case; abuse the 
plaintiff^s attorney." Whoever thus uses what is known as 
argumentuni ad homine^n^ that is an argument which 
rests, not upon the merit of the case, but the character or 
position of those engaged in it, commits this fallacy. If 
a man is accused of a crime it is no answer to say that 



XXL] MA TERIAL FALLA CIES. 1 79 

the prosecutor is as bad. If a great change in the law is 
proposed in Parhament, it is an Irrelevant Conclusion to 
argue that the proposer is not the right man to bring it 
forward. Everyone who gives advice lays himself open 
to the retort that he who preaches ought to practise, or 
that those who live in glass houses ought not to throw 
stones. Nevertheless there is no necessary connection 
between the character of the person giving advice and 
the goodness of the advice. 

The argumentum ad popuhirn is another form of 
Irrelevant Conclusion, and consists in addressing argu- 
ments to a body of people calculated to excite their feel- 
ings and prevent them from forming a dispassionate 
judgment upon the matter in hand. It is the great 
weapon of rhetoricians and demagogues. 

Petitio Principii is a familiar name, and the nature of 
the fallacy it denotes is precisely expressed in the phrase 
begging the question. Another apt name for the fallacy is 
circnlus in probanda^ or '*a circle in the proof." It con- 
sists in taking the conclusion itself as one of the premises 
of an argument. Of course the conclusion of a syllogism 
must always be contained or implied in the premises, but 
only when those premises are combined, and are dis- 
tinctly different assertions from the conclusion. Thus in 
the syllogism, 

B\sC, 

A is B, 

therefore A is C, 

the conclusion is proved by being deduced from two 
propositions, neidier of which is identical with it; but if 
the truth of one of these premises itself depends upon 
the following syllogism, 

C is B, 

A is C, 

therefore A is B, 

1 2 — g 



i8o MATERIAL FALLACIES, [less. 

it is plain that we attempt to prove a proposition by itself, 
which is as reasonable as attempting to support a body 
upon itself. It is not easy to illustrate this kind of fal- 
lacy by examples, because it usually occurs in long argu- 
ments, and especially in wordy metaphysical writings 
We are very likely to fall into it however when we employ 
a mixture of Saxon and Latin or Greek words, so as to 
appear to prove one proposition by another which is 
really the same expressed in diiferent terms, as in the 
following: "Consciousness must be immediate cognition 
of an object ; for I cannot be said really to know a thing 
unless my mind has been affected by the thing itself" 

In the use of the disjunctive syllogism this fallacy is 
likely to happen ; for by enumerating only those alterna- 
tives which favour one view and forgetting the others it is 
easy to prove anything. An instance of this occurs in the 
celebrated sophism by which some of the ancient Greek 
philosophers proved that motion was impossible. For, 
said they, a moving body must move either in the place 
where it is or the place where it is not ; now it is absurd 
that a body can be where it is not, and if it moves it can- 
not be in the place where it is; therefore it cannot move 
at all. The error arises in the assumption of a premise 
which begs the question: the fact of course is that the 
body moves between the place where it is at 07ie tnoment 
and the place where it is at the next moment, 

Jeremy Bentham however pointed out that the use 
even of a single name may imply a Petitio Principii. 
Thus in a Church assembly or synod, where a discussion 
is taking place as to whether a certain doctrine should be 
condemned, it would be a Petitio Principii to argue that 
the doctrine is heresy^ and therefore it ought to be con- 
demned. To assert that it is heresy is to beg the question, 
because every one understands by heresy a doctrine 
which is to be condemned. Similarly in Parliament a 



• XXL] ^MATERIAL FALLACIES, i8i 

bill is often opposed on the ground that it is unconstitu- 
tional and therefore ought to be rejected ; but as no 
precise definition can be given of what is or is not con- 
stitutional, it means little more than that the measure is 
distasteful to the opponent. Names which are used in 
this fallacious manner were aptly called by Bentham 
Question-begging Epithets, In like manner we beg the 
question when we oppose any change by saying that it is 
un-English. 

The Fallacy of the Consequent is better understood 
by the familiar phrase non seqtiitur. We may apply 
this name to any argument which is of so loose and 
inconsequent a character that no one can discover any 
cogency in it. It thus amounts to httle more than the 
assertion of a conclusion which has no connection with 
the premises. Prof. De Morgan gives as an example 
the following: "Episcopacy is of Scripture origin; the 
Church of England is the only episcopal Church in Eng- 
land; ergo, the Church established is the Church that 
should be supported." 

By the Fallacy of the False Cause I denote that which 
has generally been referred to by the Latin phrase non 
cazisa pro causd. In this fallacy we assume that one 
•thing is the cause of another without any sufficient 
grounds. A change in the weather is even yet attributed 
to the new moon or full moon which had occurred shortly 
before, although it has been demonstrated over and over 
again that the moon can have no such effect. In former 
centuries any plague or other public calamity which fol- 
lowed the appearance of a comet or an eclipse was 
considered to be the result of it. The Latin phrase /^j/ 
hot, ergo propter hoc (after this and therefore in conse- 
quence of this) exactly describes the character of these 
fallacious conclusions. Though we no longer dread signs . 
and omens, yet we often enough commit the fallacy; as 



i82 MATERIAL FALLACIES, [less. xxi. 

when we assume that all the prosperity of England is the 
result of the national character, forgetting that the plenti- 
ful coal in the country and its maritime position have 
contributed to our material wealth. It is no doubt equally 
fallacious to attribute no importance to national character, 
and to argue that because England has in past!: centuries 
misgoverned Ireland all the present evils of Ireland are 
due to that misgovernment. 

Lastly there is the somewhat trivial Fallacy of Many 
Questions, which is committed by those who so combine 
two or three questions into one that no true answer can 
be given to them. I cannot think of a better example 
than the vulgar pleasantry of asking, " Have you left off 
beating your mother.'*" Questions equally as unfair are 
constantly asked by barristers examining witnesses in a 
court of justice, and no one can properly be required to 
answer Yes or No to every question which may be ad- 
dressed to him. As Aristotle says, " Several questions 
put as one should be at once decomposed into their 
several parts. Only a single question admits of a single 
answer: so that neither several predicates of one subject, 
nor one predicate of several subjects, but only one predi- 
cate of one subject, ought to be affirmed or denied in a 
single answer." 

Read Prof, de Morgan's excellent and amusing Chapter 

on Fallacies, Forjnal Logic, Ch. xiii. 
Whately's remarks on Fallacies, Elements of Logic^ 

Book III., are often very original anc^ acute. 



LESSON XXII. 

THE QUANTIFICATION OF THE PREDICATE. 

The syllogism has been explained in the preceding three 
lessons almost exactly in the form in which it has been 
taught for more than two thousand years. Just as Geo- 
metry has been taught in the way and order first adopted 
by the ancient Greek writer Euclid, so Logic has been 
taught nearly as Aristotle taught it about the year 335 B.C. 

But within the last few years teachers have at last 
come to the conclusion in England that Euclid's ideas of 
Geometry are not as perfect as could be desired. During 
the last 30 or 40 years also it has been gradually made 
apparent that Aristotle's syllogism is not an absolutely 
perfect system of logical deduction. In fact, certain 
eminent writers, especially Sir William Hamilton, Pro- 
fessor De Morgan, Archbishop Thomson and Dr Boole, 
have shewn that we need to make improvements from the 
very basis of the science. 

This reform in Logic is called by the somewhat mys- 
terious name of the quantification of the predicate, but 
the reader who has found no insuperable difficulty in 
the preceding lessons need not fear one here. To quan- 
tify the predicate is siinply to state whether the whole or 
the pari only of the predicate agrees with or differsfrcns. 
the subject. In this proposition, 

" All m.etals are elements," 



THE QUANTIFICATION [less, 

the subject is quantified, but the predicate is not; we 
know that all metals are elements, but the proposition 
does not distinctly assert whether metais make the v/hole 
of the elements or not. In the quantified proposition 

" All metals are some elements," 
the little word some expresses clearly that in reality the 
metals form only a part of the elements. Aristotle avoid- 
ed the use of any mark of quantity by assuming, as we 
have seen, that all affirmative propositions have a par- 
ticular predicate, like the example just given ; and that 
only negative propositions have a distributed or universal 
predicate. The fact however is that he was entirely in 
error, and thus excluded from his system an infinite 
number of affirmative propositions which are universal 
in both terms. It is true that — 
"All equilateral triangles are all equiangular triangles," 

but this proposition could not have appeared in his system 
except in the mutilated form — 

"Ail equilateral triangles are equiangular." 
Such a proposition as 

"London is the capital of England,'* 
or " Iron is the cheapest metal," 

had no proper place whatever in his syllogism, since both 
terms are singular and identical with each other, and 
both are accordingly universal. 

As soon as we allow the quantity of the predicate to 
be stated the forms of reasoning become much simplified. 
We may first consider the process of conversion. In our 
lesson on the subject it was necessary to distinguish be- 
tween conversion by limitation and simple conversion. 
But now one single process of simple conversion is suffi- 
cient for all kinds of propositions. Thus the quantified 
proposition of the form A, 

"All metals are some elements," 



XXII.] OF THE PREDICATE, 18-5 

is simply converted into 

"Some elements are all metals, ' 
The particular affirmative proposition 

" Some metals are some brittle substances'* 
becomes by mere transposition of terms 

" Some brittle substances are some metals." 
The particular negative proposition 

" Some men are not (any) trustworthy persons " 
is also converted simply into 

" Not any trustworthy persons are some men,'' 
though the result may appear less satisfactory in this form 
than in the affirmative form, as follows, 

" Some men are some not-trustworthy persons," 
converted simply into 

" Some not -trustworthy persons are some men." 

The universal negative proposition E is converted 
simply as before, and finally we have a new affirmative 
proposition universal both in subject and predicate ; as in 

"All equilateral triangles are all equiangular triangles," 

which may obviously be converted simply into 

"All equiangular triangles are all equilateral triangles." 

This doubly universal affirmative proposition is ol 
most frequent occurrence; as in the case of all definitions 
and singular propositions ; I may give as instances 
"Honesty is the best policy," "The greatest truths are 
the simplest truths," "Virtue alone is happiness below," 
" Self-exaltation is the fool's paradise." 

When affirmative propositions are expressed in the 
quantified form all immediate inferences can be readily 
drawn from them by this one rule, that whateuer we do 
with one term we should do with the other ter7n. Thus 
from the doubly universal proposition, "Honesty is the 
best policy," we infer that "what is not the best policy is 



1 86 THE QUANTIFICATION [LESS 

not honesty," and also " what is not honesty is not the best 
policy." From this proposition in fact we can draw two 
contrapositives ; but the reader will carefully remember 
that from the ordinary unquantified proposition A we 
can only draw one contrapositive (see p. 84). Thus if 
"metals are elements" we must not say that "what are 
not metals are not elements." But if we quantify the 
predicate thus, "All metals are some elements," we may 
infer that " what are not metals are not some elements.'* 
Immediate inference by added determinant and complex 
conception can also be applied in either direction to 
quantified propositions without fear of the errors noticed 
in pp. 86-7. 

It is clear that in admitting the mark of quantity before 
the predicate we shall double the number of propositions 
which must be admitted into the syllogism, because the 
predicate of each of the four propositions A, E, I, may 
be either universal or particular. Thus we arrive at a list 
of eight conceivable kinds of propositions, which are 
stated in the following table. 

U MIX is all K. \ 

I Some X is some Y, I Affirmative 

A All X is some K f propositions. 

Y Some X is all Y. J 



E No X is (any) K 

w Some X is not some K 

tj No X is some K 

Some X is no K 



Negative 
propositions. 



The letters X and Y are used to stand for any subject 
and predicate respectively, and the reader by substituting 
various terms can easily make propositions of each kind. 
The symbolic letters on the left-hand side were proposed 
by Archbishop Thomson as a convenient mode of reter- 



XXII.] OF THE PREDICATE. 187 

jring to each of the eight propositions, and are very 
suitably chosen. The doubly universal affirmative pro- 
position is called U ; the simple converse of A is called 
Y; the Greek letter t| {Eta^ e) is applied to the proposi- 
tion obtained by changing the universal predicate of E 
into a particular predicate ; and the Greek w {Omega^ 0) 
is applied to the proposition similarly determined from 0. 
All these eight propositions are employed by Sir W. Ha- 
milton, but Archbishop Thomson considers that two of 
them, 11 and «, are never really used. It is remarkable 
that a complete table of the above eight propositions was 
given by Mr George Bentham in a work called Outline 
of a New System of Logic ^ published in 1827, several 
years previous to the earliest of the logical publications of 
Sir W. Hamilton. But Mr Bentham considered that some 
of the propositions are hardly to be distinguished from 
others; as Y from A, of which it is the simple converse; or 
Tj from 0. 

The employment even of the additional two proposi- 
tions U and Y introduced by Thomson much extends 
the list of possible syllogisms, making them altogether 62 
in number, without counting the fourth figure, which is 
not employed by Hamilton and Thomson. When the 
whole eight propositions are admitted into use we are 
obliged to extend the list of possible syllogisms so as to 
contain 12 affirmative and 24 negative moods in each of 
the three first figures. The whole of these moods are 
conveniently stated in the table on^the next page, given by 
Archbishop Thomson at p. 188 of his Laws of Thought, 

Sir W. Hamilton also devised a curious system of 
notation for exhibiting all the moods of the syllogism in a 
clear manner. He always employed the letter J/ to denote 
the middle term of the syllogism, and the two letters C 
and r (the Greek capital letter Gamma) for the two 
terms appearing in the conclusion. The copula of the 



i88 



THE QUANTIFICATION 



[less. 



Table of Moods of the Syllogism, 





First Figure. 


Second Fig. 


Third Figure. 




Affirm. 


Neg. 


Affirm. 


Neg. 


Affirm. 


Neg. 


1 


UUU 


EUE 
UEE 


UUU 


EUE 

UEE 


UUU 


EUE 
UEE 


• • 

11 


AYI 


7;Y 0) 
AOft) 


YYI 


OYft) 
YOft) 


AAI 


77 A ft) 
A 77 ft) 


iii 


AAA 


y)Kr) 


YAA 


OA^ 


AYA 


1^1 


iv 


YYY 


OYO 


AYY 


Y„ 
,;Y0 


YAY 


AOrj 

OAO 


V 


All 


YOO 

T; I ft) 


YII 


AOO 
Olft) 


All 


Yi;0 

7; I ft) 


vi 


lYI 


Aft) ft) 
ft) Yft) 


lYI 


Y ft) ft) 
ft) Y ft) 


lAI 


A ft) ft) 
ft) A ft) 


vii 


UYY 


lOft) 
EYO 


UYY 


I Oft) 
EYO 


UAY 


1 77 ft) 

EAO 


viii 


AUA 


U 00 


YUA 


UOO 
OUt? 


AUA 


U7;0 


ix 


UAA 


AEt; 
EAE 


UAA 


YEt? 
EAE 


UYA 


AEt} 
EYE 


X 


YUY 


OUO 
YEE 


AUY 


TyUO 

AEE 


Y UY 


VOrj 

OUO 

YEE 


xi 


UII 


EIO 


UII 


EIO 


UII 


EIO 


• • 

Xll 


lUI 


U ft) ft) j 
ft) U ft) 

IE, 1 


lUI 


U ft) ft) 
ft) U ft) 

IE;? 


lUI 


U ft) ft) 
ft) U ft) 

IEf7 



proposition was indicated by a line thickened towards 

the subject ; thus C i^^ J/ means that " Cisi^," 

To indicate the quantity of the terms Hamilton inserted a 




XXJI.J OF THE PREDICATE, 189 

colon (:) between the term and the copula when the 
quantity is universal, and a comma (,) when the quantity 
is particular. Thus we readily express the following 
affii'mative propositions. 

, M All Cs are some J/'s (A) 
: M All C's are all M's (U) 

, M Some C's are some APs (I) 

and so on. Any affirmative proposition can be converted 
into the corresponding negative proposition by drawing a 
stroke through the line denoting the copula, as in the 
following — 

\ M No C is any M /'E) 

: M Some C is not any M (0) 
, M Some C is not some M (w) 

Any syllogism can be represented by placing M the 
middle term in the centre and connecting it on each side 
with the other terms. The copula representing the con- 
clusion can then be placed below ; Barbara is expressed 
as follows — 




The negative mood Celarent is similarly — 

|sB52^^a:i^, "^'^ -p 




^as^^^^ 



Cesare in the second figure is thus represented- 



Sir W. Hamilton also proposed a new law or suprems 
Janon of the syllogism by which the validity of all forms 



I90 THE QUANTIFICATION [less. 

of the 'Syllogism might be tested. This was stated in thp 
following words : "What worse relation of subject and 
predicate subsists between either of two terms and a 
common third term, with which both are related, and one 
at least positively so — that relation subsists between these 
two terms themselves." 

By a worse relation^ Sir William means that a negative 
relation is worse than an affirmative and a particular than 
a universal. This canon thus expresses the rules that if 
there be a negative premise the conclusion must be nega- 
tive, and if there be a particular premise the conclusion 
must be particular. Special canons were also developed 
for each of the three figures, but in thus rendering the 
system complex the advantages of the quantified form of 
proposition seem to be lost. 

Prof. De Morgan also discovered the advantages of 
the quantified predicate, and invented a system differing 
greatly from that of Sir W. Hamilton. It is fully ex- 
plained in his Formal Logic, The Syllabus of a new 
System of Logic, and various important memoirs on the 
Syllogism in the Transactions of the Cambridge Philo- 
sophical Society. In these works is also given a com- 
plete explanation of the *' Numerically Definite Syllogism.** 
Mr De Morgan pointed out that two particular premises 
may often give a valid conclusion provided that the 
actual quantities of the two terms are stated, and when 
added together exceed the quantity of the middle term. 
Thus if the majority of a public meeting vote for the first 
resolution, and a majority also vote for the second, it 
follows necessarily that some who voted for the first voted 
also for the second. The two majorities added together 
exceed the whole number of the meeting, so that they 
could not consist of entirely difierent people. They may 
indeed consist of exactly the same people ; but all that 
we can deduce from the premises is that the excess of the 



XXI r.] OF THE PREDICATE, 191 

two majorities added together over the number of the 
jneeting must have voted in favour of each resolution. 
This kmd of inference has by Sir W. Hamilton been 
baid to depend on ultra-total distribution ; and the name 
of Plurative Propositions has been proposed for all those 
which give a distinct idea of the fraction or number of the 
subject involved in the assertion. 

T. Spencer Baynes, Essay on the new Analytic oj 

Logical Forms; Edinburgh, 1850. 
Frof. Bo wen's Treatise on Logic or the Laws of Pure 

Thought, Cambridge, U. S. 1866 (Trubner and 

Co.) gives a full and excellent account of Hamilton's 

Log.c. 



LESSON XXIII. 
BOOLE'S SYSTEM OF LOGIC. 

It would not in the least be possible to give in an ele- 
mentary work a notion of the system of indirect inference 
first discovered by the late Dr Boole, the Professor of 
Mathematics at the Queen's College, Cork. This system 
was founded as mentioned in the last lesson upon the 
Quantification of the Predicate, but Dr Boole regarded 
Logic as a branch of Mathematics, and believed that he 
could arrive at every possible inference by the principles 
of algebra. The process as actually employed by him 
is very obscure and difficult ; and hardly any attempt to 
introduce it into elementary text-books of Logic has yet 
been made. 

I have been able to arrive at exactly the same results 



192 BOOLE'S SYSTEM OF LOGIC. [less. 

as Dr Boole without the use of any mathematics; and 
though the very simple process which I am going to 
describe can hardly be said to be strictly Dr Boole's 
logic, it is yet very similar to it and can prove everything 
that Dr Boole proved. This Method of Indirect Inference 
is founded upon the three primary Laws of Thought 
stated in Lesson xiv., and the reader who may have 
thought them mere useless truisms will perhaps be sur- 
prised to find how extensive and elegant a system of 
deduction may be derived from them. 

The law of excluded middle enables us to assert that 
anything must either have a given quality or must have it 
not. Thus if iron be the thing, and combustibility the 
quality, anyone must see that 

"Iron is either combustible or incombustible." 

This division of alternatives may be repeated as often 
as we like. Thus let Book be the class of things to be di- 
vided, and English and Scientific two qualities. Then any 
book must be either English or not English; again an 
English book must be either Scientific or not Scientific, 
and the same may be said of books which are not English. 
Thus we can at once divide books into four classes — 

Books, English and Scientific. 
Books, English and not-Scientific. 
Books, not-English and Scientific. 
Books, not-English and not-Scientific. 

This is what we may call an exhaustive division of the 
class Books; for there is no possible book which does 
not fall into one division or other of these four, on 
account of the simple reason, that if it does not fall into 
any of the three first it must fall into the last. The pro- 
cess can be repeated without end, as long as any new 
circumstance can be suggested as the ground of division. 
Thus we might divide each class again according as the 



XXIII.] BOOLE'S SYSTEM OF LOGIC. 193 

books are octavo or not octavo, bound or unbound, pub- 
lished in London or elsewhere, and so on. We shall call 
this process of twofold division, which is really the pro- 
cess of Dichotomy mentioned in p. 107, the development 
of a term, because it enables us always to develope the 
utmost number of alternatives which need be considered. 
As a general rule it is not likely that all the alterna- 
tives thus unfolded or developed can exist, and the nexf 
point is to ascertain how many do or may exist. The La\^ 
of Contradiction asserts that nothing can combine con- 
tradictory attributes or qualities, and if we meet with any 
term which is thus self-contradictory we are authorized at 
once to strike it out of the list. Now consider our old 
example of a syllogism : 

Iron is a metal ; 

All metals are elements ; 

Therefore iron i-s an element. 

We can readily prove this conclusion by the indirect 
method. For if we develope the term iron, we have four 
alternatives ; thus — 

Iron, metal, element. 

Iron, metal, not-element. 

Iron, not-metal, element. 

Iron, not-metal, not-element. 

But if we compare each of these alternatives with the 
premises of the syllogism, it will be apparent that several 
of them are incapable of existing. Iron, we are informed, 
is a metal. Hence no class of things ''iron, not-metal" 
can exist. Thus we are enabled by the first premise to 
strike out both of the last two alternatives which combine 
iron and not-metal. The second alternative, again, com- 
bines metal and not-element ; but as the second premise 
informs us that "all metals are elements," it must be 
struck out. There remains, then, only one alternative 

13 



194 BOOLE'S SYSTEM OF LOGIC, [less. 

which is capable of existing if the premises be tnie, and as 
there cannot conceivably be more alternatives than those 
considered, it follows demonstratively that iron occurs 
only in combination with the qualities of metal and ele- 
ment, or, in brief, that it is an element. 

We can, however, prove not only the ordinary syllo- 
gistic conclusion, but any other conclusion which can be 
drawn from the same premises ; the syllogistic conclusion 
is in fact only one out of many which can usually be ob- 
tained from given premises. Suppose, for instance, that 
we wish to know what is the nature of the term or class 
7iot-eleinent^ so far as we can learn it from the premises 
just considered. We can develope the alternatives of this 
term, just as we did those of iron, and get the following— 

Not-element, iron, metal. 
Not-element, iron, not-metal. 
Not-element, not-iron, metal. 
Not-element, not-iron, not-metal. 

Compare these combinations as before with the premises. 
The first it is easily seen cannot exist, because all metals 
are elements ; for the same reason the third cannot exist ; 
the second is likewise excluded, because iron is a metal 
and cannot exist in combination with the qualities of not- 
metal. Hence there remains only one combination to 
represent the class desired — namely, 

Not-element, not-iron, not-metal. 

Thus we learn from the premises that every not-ele 
ment is not a metal and is not iron. 

As another example of this kind of deductive process 
I will take a case of the Disjunctive Syllogism, in the ne* 
gative mood, as follows : — 

A fungus is either plant or animal, 
A fungus is not an animal ; 
Therefore it is a plant 



XXiii.J BOOLE'S SYSTEM OF LOGIC, 195 

Now if we develope all the possible ways in which 
fungus, plant and animal can be combined together, we 
obtain for the term fungus — 

(i) Fungus, plant, animal. 

(2) Fungus, plant, not-animal. 

(3) Fungus, not-plant, animal. 

(4) Fungus, not-plant, not-animal. 

Of these however the 4th cannot exist because by 
the premise a fungus must be a plant, or if not a plant an 
animal. The ist and 3rd again cannot exist because thfe 
minor premise informs us that a fungus is not an animal 
There remains then only the second combination. 

Fungus, plant, not-animal, 

from which we learn the syllogistic conclusion that 
" a fungus is a plant." 

The chief excellence of this mode of deduction consists 
in the fact that it is not restricted to any definite series 
of forms like the syllogism, but is applicable, without any 
additional rules, to all kinds of propositions or problems 
which can be conceived and stated. There may be any 
number of premises, and they may contain any number of 
terms ; all we have to do to obtain any possible inference 
is to develope the term required into all its alternatives 
and then to examine how many of these agree with the pre- 
mises. What remain after this examination necessarily 
form the description of the term. The only inconvenience 
of the method is that, as the number of terms increases, 
the number of alternatives to be examined increases very 
rapidly, and it soon becomes tedious to write them all out. 
This work may be abbreviated if we substitute single 
letters to stand for the terms, somewhat as in algebra; 
thus we may take^,ij', C, D^ &c., to stand for the affirm- 
ative terms, and ^, ^, ^, d, &c., for the corresponding nega- 
tive ones. Let us take as a first example the premises — 

13—2 



196 BOOLE'S SYSTEM OF LOGIC, [less. 

Organic substance is either vegetable or animal 
Vegetable substance consists mainly of carbon, hydrogeiij 

and nitrogen. 
Animal substance consists mainly of carbon, hydrogen, 

and nitrogen. 

It would take a long time to write out all the combi- 
nations of the four terms occurring in the above; but if 
We substitute letters as follows — 

A = organic substance, 

^ = vegetable substance, 

C= animal substance, 

D — consisting mainly of carbon, hydrogen, and 
nitrogen, 
we can readily represent all the combinations which can 
belong to the term A, 



(I) 


ABCD 


AbCD 


(s) 


(2) 


ABCd 


AbCd 


(6) 


(3) 


ABcD 


AbcD 


(7) 


(4) 


ABcd 


Abed 


(8) 


Now the pn 


smises amount 


to the J 


jtatei 



A must be either B ox C, 
B must be D^ 
C must be D, 

The combinations (7) and (8) are inconsistent with the 
first premise ; the combinations (2) and (4) with the second 
premise; and (6) is inconsistent with the third premise. 
There remain only, 

ABCD 
ABc'D 
AbCD, 

Whence we learn at once that "organic substance {A) 
always consists mainly of carbon, hydrogen and nitrogen^" 



XXIII.] BOOLE? S SYSTEM OF LOGIC. 197 

because it always occurs in connexion with D. The reader 
may perhaps notice that the term A BCD impHes that or- 
ganic substance may be both vegetable {B) and animal {C\ 
If the first premise be interpreted as meaning that this is 
not possible, of course this combination should also be 
struck out. It is an unsettled point whether the alter- 
natives of a disjunctive proposition can coexist or not 
(see p. 166)5 but I much prefer the opinion that they 
can; and as a matter of fact it is quite likely that there 
exist very simple kinds of living beings, which cannot be 
distinctly asserted to be vegetable only or animal only, 
but partake of the nature of each. 

As a more complicated problem to shew the powers of 
this system, let us consider the premises which were 
treated by Dr Boole in his Laws of Thought^ p. 125, as 
follows : 

" Similar figures consist of all whose corresponding 
angles are equal, and whose corresponding sides are 
proportional. 

Triangles whose corresponding angles are equal have 
their corresponding sides proportional ; and vice versa. 

Triangles whose corresponding sides are proportional 
have their corresponding angles equal." 

Now if we take our symbol letters as follows : 

A = similar figure, 

^ = triangle, : 

C= having corresponding angles equal, 

Z) = having corresponding sides proportional, 

the premises will be seen to amount to the statements that 

A is identical with CD^ 
and that 

BC is identical with BD; 
hi other words, all A\ ought to be CU^, CD's ought to 



198 BOOLE'S SYSTEM OF LOGIC. [less. 

be A% all BCs ought to be BD's and all BHs ought to 
be BCs. 

The possible combinations in which the letters may be 
united are i6 in number and are shewn in the following 
table ; 



A BCD 


aBCD 


ABCd 


aBCd 


ABcD 


aBcD 


ABc d 


aB cd 


AbCD 


abCD 


AbCd 


ab Cd 


AbcD 


ab c D 


A bed 


abed 



Comparing each of these combinations with the premise! 
we see that ABCdy ABcD^ ABcd, and others, are to b6> 
struck out because every A is also to be CD. The com- 
binations aBCD and abCD are struck out because ever> 
CD should also be A» Again, aBCd is inconsistent with 
the condition that every BC is also to be BD\ and if 
the reader carefully follows out the same process of ex- 
amination, there will remain only six combinations, which 
agree with all the premises, thus — 

I ABCD aBed 

^ AbCD abCd 

\ abeD 

\ abed 

rom these combinations we can draw any description 
ve like of the classes of things agreeing with the premises. 
The class A or similar figures is represented by only two 
combinations or alternatives ; the negative class a or 
dissimilar figures, by four combinations, whence we may 

raw the following conclusion: "Dissimilar figures con- 
sist of all triangles which have not their corresponding 
^gles equal, and sides proportional {ciBed\ and of all 






\ 



XXIII.] BOOLE'S SYSTEM OF LOGIC. 199 

figures, not being triangles, which have either their angles 
equal and sides not proportional {abCd), or their cor- 
responding sides proportional and angles not equal 
{abcD)y or neither their corresponding angles equal nor 
corresponding sides proportional iabcd)J^ 

In performing this method of inference it is soon seen 
to proceed in a very simple mechanical manner, and the 
only inconvenience is the large number of alternatives or 
combinations to be examined. I have, therefore, devised 
several modes by which the labour can be decreased; 
the simplest of these consists in engraving the series 
of 16 combinations on the opposite page, which occur 
over and over again in problems, with larger and smaller 
sets, upon a common writing slate, so that the excluded 
ones may be readily struck out with a common slate 
pencil, and yet the series may be employed again for any 
Tuture logical question. A second device, which I have 
called the "Logical abacus," is constructed by printing the 
letters upon slips of wood furnished with pins, contrived 
so that any part or class of the combinations can be 
picked out mechanically with very little trouble ; and a 
logical problem is thus solved by the hand, rather than 
by the head. More recently however I have reduced th^ 
system to a completely me * .nica"* form, and have thui 
embodied the whole of the indirect process of inference 
in what may be called a Logical Machine. In the front 
of the machine are seen certain moveable wooden rods 
carrying the set of 16 combinations of letters which are 
seen on the preceding page. At the foot are 21 keys like 
those of a piano ; eight keys towards the left hand are 
marked with the letters Ay a^ B, b, C, c, D, dy and are 
intended to represent these terms when occurring in the 
subject of a proposition. Eight other keys towards the 
right hand represent the same letters or terms when oc- 
curring in the predicate. The copula of a proposition ia 



200 BOOLE S SYSTEM OF LOGIC, [LEsa 

represented by a key in the middle of the series ; the full 
stop by one to the extreme right, while there are two other 
keys which serve for the disjunctive conjunction or^ ac- 
cording as it occurs in subject or predicate. Nov/ if the 
letters be taken to stand for the terms of a syllogism or 
any other logical argument, and the keys of the instru- 
ment be pressed exactly in the order corresponding to the 
words of the premises, the i6 combinations will" be so 
selected and arranged thereby that at the end only the 
possible combinations will remain m view. Any question 
can then be asked of the machine, and an infallible answer 
will be obtained from the combinations remaining. The 
internal construction of the machine is such, therefore, as 
actually to perform the work of inference which, in Dr 
Boole's system, was performed by a very complicated 
mathematical calculation. It should be added, that there 
is one remaining key to the extreme left which lias the 
effect of obliterating all previous operations and restoring 
all the combinations to their original place, so that the 
machine is then ready for the performance of any new 
problem. 

i An account of this logical machine may be found in 
the Proceedings of the Royal Society for Jan. 2Qth, 1870, 
the machine having on that day been exhibited in action to 
^he Fellows of the Society. The principles of the method 
(i)f inference here described are more completely stated in 
^he Substitution of Similars'^ , and the Pure Logic '\, "which. 
1 published in the years 1869 and 1864. I may add, that 
the first-named of these works contains certain view^s as 
to the real nature of the process of inference which I do 

* The Substitution of Similars^ the true Principle of Reason- 
ingf derived from a modification of Aristotle's Dictufn» Mac- 
millan and Co. 1869. 

t Pure Logic^ or the Logic of Quality apart from Quantity ^^c* 
Edward Stanford, Charing Cross. 



XXIII.] BOOLE'S SYSTEM OF LOGIC, 201 

n:>t think it desirable to introduce into an elementary work 
like the present, on account of their speculative character. 
The process of inference, on the other hand, which I have 
derived from Boole's system is of so self-evident a charac- 
ter, and is so clearly proved to be true by its reduction to 
a mechanical form, that I do not hesitate to bring it to the 
reader's notice. 

George Boole, Mathematical Analysis of Logic y 1847, 
An Investigation of the Laws of Thought, Londor 
Walton and Maberly, 1854. 



LESSON XXIV. 

ON METHOD, ANALYSIS AND SYNTHESIS. 

It has been held by many writers on Logic that, in addi- 
tion to the three parts of logical doctrine which treat 
successively of Terms, Propositions and Syllogisms, there 
was a fourth part, which treats of method. Just as the 
doctrine of Judgment considers the arranging of terms 
and their combination into Propositions, and the doc- 
trine of Syllogism considers the arranging of propositions 
that they may form arguments, so there should in like 
manner be a fourth part, called Method, which should 
govern the arrangement of syllogisms and their combina- 
tion into a complete discourse. Method is accordingly 
defined as consisting in such a dispositio7t of the parts oj 
a discourse that the whole may be most easily intelligible. 
The celebrated Peter Ramus, who perished in the 
massacre of St Bartholomew, first proposed to make 
method in this manner a part of the science of Logic ; but 



202 ON METHOD, ANALYSIS [l£SS-. 

it may well be doubted whether any definite set of rules 
or principles can be given to guide us in the arrangement 
of "arguments. Every different discourse must consist of 
arguments arranged in accordance with the peculiar nature 
of the subject ; and no general rules can be given for treat- 
ing things which are infinitely various in the mode of treat- 
ment required. Accordingly the supposed general rules 
of method are no better than truisms, that is, they tell us 
nothing more than we must be supposed to know before- 
hand. Thus, we are instructed in composing any dis- 
course to be careful that — 

1. Nothing should be wanting or redundant. 

2. The separate parts should agree with each other. 

3. Nothing should be treated unless it is suitable to 
the subject or purpose. 

4. The separate parts should be connected by suit- 
able transitions. 

But it is evident that the whole difficulty consists in 
deciding what is wanting or redundant, suitable or con- 
sistent. Rules of this kind simply tell us to do what we 
ought to do, without defining what that is. 
' I There exist nevertheless certain general modes of 
tjreating any subject which can be clearly distinguished, 
and should be well understood by the logical student. 
Logic cannot teach him exactly how and when to use 
each kind of method, but it can teach him the natures 
and powers of the methods, so that he will be more likely 
to use them rightly. We must distinguish, 

1. The method of discovery, 

2. The method of instruction. 

The method of discovery is employed in the acquisi- 
tion of knowledge, and really consists in those processes 
of inference and induction, by which general truths are 
ascertained from the collection and examination of par- 



XXIV.] AND SYNTHESIS. 203 

ticular facts. This method will be the subject of most of 
our remaining Lessons. The second method only applies 
when knowledge has already been acquired and express- 
ed in the form of general laws, rules, principles or truths, 
so that we have only to make ourselv^^^ acquainted with 
these and observe the due mode 01 applying them to 
particular cases, in order to possess a complete acquaint- 
ance with the subject. 

A student, for example, in learning Latin, Greek, 
French, German, or any well-known language, receives a 
complete Grammar and Syntax setting forth the whole of 
the principles, rules and nature of the language. He 
receives these instructions, and takes them to be true on 
the authority of the teacher, or the writer of the book; 
and after rendering them familiar to his mind he has 
nothing to do but to combine and apply the rules in read- 
ing or composing the language. He follows, in short, 
the method of Instruction. But this is an entirely differ- 
ent and opposite process to that which the scholar must 
pursue who has received some writings in an unknown 
language, and is endeavouring to make out the alpha- 
bet, words, grammar, and syntax of the language. He 
possesses not the laws of grammar, but words and sen- 
ten ce§ obeying those laws, and he has to detect the 
laws if possible by observing their effects on the written 
language. He pursues, in short, the method of discovery 
consisting in a tedious comparison of letters, words, and 
phrases, such as shall disclose the more frequent combi- 
nations and forms in which they occur. The process 
would be a strictly inductive one, such as I shall partially 
exemplify in the Lessons on Induction ; but it is far more 
difficult than the method of Instruction, and depends to a 
great extent on the happy use of conjecture and hypothesis, 
which demands a certain skill and inventive ability. 

Exactly the same may be said of the investigation of 



J 



204 ON METHOD, ANALYSIS [less 

natural things and events. The principles of mechanics^ 
of the lever, inclined plane, and other Mechanical Powers, 
or the Laws of Motion, seem comparatively simple and 
obvious as explained to us in books of instruction. But 
the early philosophers did not possess such books; they 
had only the Book of Nature, in which is set forth not 
the laws but the results of the laws, and it was only 
after the most patient and skilful investigation, and after 
hundreds of mistakes, that those laws were ascertained. 
It is very easy now to understand the Copernican system 
of Astronomy, which represents the planets as revolving 
round the sun in orbits of various magnitude. Once know- 
ing the theory we can readily see why the planets have 
such various movements and positions, and why they 
som.etimes stand still ; it is easy to see, too, why in ad- 
dition to their own proper motions they all go round the 
earth apparently every day in consequence of the earth's 
diurnal rotation. But all these changes were exceedingly 
puzzling to the ancients, who regarded the earth as stand- 
ing still. 

The metliod of discovery thus begins with facts ap- 
parent to the senses, and has the difficult task of detecting 
thQse universal laws or general principles which can only 
be comprehended by intellect. It has been aptly said 
that the method of discovery thus proceeds from things 
better known to us, or our senses {nobis notiord), to those 
which are more simple or better known in natii7'e {notiora 
natures). The method of Instruction proceeds in the 
opposite direction, beginning with the things notiora 
natures, and proceeding to show or explain the things 
nobis notiora. The difference is almost like that between 
hiding and seeking. He who has hidden a thing knows 
where to find it; but this is not the position of a discoverer, 
who has no clue except such as he may meet in his own 
diligent and sagacious search. 



XXIV.] AND SYNTHESIS. 205 

Closely corresponding to the distinction between the 
methods of Discovery and Instruction is that between 
the methods of Analysis and Syntliesis. It is very im« 
portant indeed that the reader should clearly apprehend 
the meanings of these terms in their several appHcations. 
Analysis is the process of separating a whole into its 
parts, and synthesis the combination of parts into a 
whole. The analytical chemist, who receives a piece of 
mineral for examination, may be able to separate com- 
pletely the several chemical elements of which it is 
composed and ascertain their nature and comparative 
quantities ; this is chemical analysis. In other cases the 
chemist mixes together carefully weighed quantities of 
certain simple substances and combines them into a new 
compound substance ; this is chemical synthesis. Logical 
analysis and synthesis must not be confused with the 
physical actions, but they are nevertheless actions of 
mind of an analogous character. 

In logical sjoitliesis we begin with the. simplest possibles- 
notions or ideas, and combine them together. We havt^' 
the best possible example in the elements of Geometryvv 
In Euclid we begin with certain simple notions of point^^/ 
straight lines, angles, right angles, circles, &c. Puttin^^ 
together three straight lines we make a triangle ; joinin-,' 
to this the notion of a right-angle, we form the notion <, 
a right-angled triangle. Joining four other equal lines j 
right angles to each other we gain the idea of a squar^j 
and if we then conceive such a square to be formed upcj 
each of the sides of a right-angled triangle, and reasG^ 
Irom the necessary qualities of these figures, we discovc 
that the two squares upon the sides containing the rigL 
angle must together be exactly equal to the square upo, 
the third side, as shewn in the 47 th Proposition o 
Euclid's first book. This is a perfect instance of com 
bining simple ideas into more complex ones. \ 



2o6 ON METHOD, ANALYSIS [lesS: 

We have often, however, in Geometry to pursue thd 
opposite course of Analysis. A complicated geometrical 
figure may be given to us, and we may have, in order to 
prove the properties which it possesses, to resolve it into 
its separate parts, and to consider the properties of those 
parts each distinct from the others. 

A similar distinction between the analytical and syn- 
thetic methods can be traced throughout the natural 
sciences. By keeping exact registers of the appearance 
and changes of the weather we may readily acquire an 
immense collection of facts, each such recorded fact 
implying a multitude of different circumstances occurring 
together. Thus in any storm or shower of rain we have 
to consider the direction and force of the wind ; the tem- 
perature and moistness of the air ; the height and forms of 
the clouds; the quantity of rain which falls, or the light- 
ning and thunder which occur with it. If we proceed by 
Q analysis only to explain the changes of the weather we 
pphould have to try resolving each storm or change of 
inweather into its separate circumstances, and comparing 
ach with every other to discover what circumstances 
paisually go together. We might thus ascertain no doubt 
thc'ith considerable certainty what kinds of clouds, and 
be hat changes of the wind, temperature, moisture, &:c., 
tha?ually precede any kind of storm, and we might even in 
^^//ne give some imperfect explanation of what takes place 
whi the atmosphere. 

^at I^^t we might also apply with advantage the syn- 
Qppctical method. By previous chemical investigations we 
na{^^^ that the atmosphere consists mainly of the two 
nol^^^ gases, oxygen and nitrogen, with the vapour of 
liijdLiQx, the latter being very variable in quantity. We 
-wb^-^ try experimentally what takes place when portions 
wh^ such air of various degrees of moistness are com- 
dili'essed or allowed to expand, or are mixed together, as 



XXIV.] AND SYNTHESIS. 207 

often happens in the atmosphere. It is thus discovered 
that whenever moist air is allowed to expand cloud 
is produced, and it may be drops of rain. Dr Hut- 
ton, too, found that whenever cold moist air is mixed 
with warm moist air cloud is again produced. We can 
safely argue from such small experiments to what takes 
place in the atmosphere. Putting together synthetically, 
from the sciences of chemistry, mechanics, and electricity, 
all that we know of air, wind, cloud and lightning, we are 
able to explain what takes place in a thunder-storm far 
more completely than we could do by merely observing 
directly what happens in the storm. We are here how- 
ever anticipating the methods of inductive investigation, 
which we must consider in the following lessons. It will 
appear that Induction is equivalent to analysis, and that 
the deductive kinds of reasoning which we have treated 
in prior lessons are of a synthetic character. 

It has been said that the synthetic method usually 
corresponds to the method of instruction and the analytic^ 
method to that of discovery. But it may be possible tc 
discover new truths by synthesis and to teach old onei^ 
by analysis. Sir John Herschel in his well-known Ouc, 
Imes of Astronomy partially adopts the analytic method' 
he supposes a spectator in the first place to survey th 
appearances of the heavenly bodies and the surface ' 
the earth, and to seek an explanation ; he then leac! 
him through a course of arguments to show that thei 
appearances really indicate the rotundity of the earth, 5 
revolution about its own axis and round the sun, and '* 
subordinate position as one of the smaller planets of t) 
solar system. Mr Norman l^ockyQr^s Elejnentary Lesso', 
in Astronomy is a clear example of the synthetic methc 
of instruction ; for he commences by describing the sur 
the centre of the system, and successively adds the plane' 
and other members of the system, until at last we ha\ 



■■■^ 



2o8 ON METHOD, ANALYSIS [less. 

the complete picture ; and the reader who has temporarily 
received everything on the writer's authority, sees that 
the description corresponds with the truth. Each method, 
it must be allowed, has its own advantages. 

It rfiust be carefully observed that the meaning of 
analysis, and therefore that of synthesis, varies according 
as we look to the intension or extension of terms. To 
divide or analyse a class of things in extension I must add 
a quality or difference. Thus I divide the class organis^n 
when I add the quality vegetable^ and separate vegetable 
organism from what is not vegetable. Analysis in exten- 
sion is therefore the same process as synthesis in inten- 
sion ; and vice versa^ whenever I separate or analyse a 
group of qualities each part belongs to a larger class of 
things in extension. When I analyse the notion vegetable 
organism, and regard the notion organism apart from 
vegetable, it is apparent that I really add the whole class 
G of animal organisms to the class I am considering — so 
pith at analysis in intension is synthesis in extension. The 
inreader who has well considered the contents of Lessons 
i^. and Xli. v/ill probably see that this connection of the 
pawo processes is only a re-statement of the law, (p. 40), 
thchat "as the intension of a term is increased the extension 
be > decreased." 

tha To express the difference between knowledge derived 
^^//sductively and that obtained inductively the Latin 
whihrases a priori and a posteriori are often used. By 
nat priori reasoning we mean argument based on truths 
oppeviously known ; A posteriori reasoning, on the contrary, 
«^/'oceeds to infer from the consequences of a general 
^^Aith what that general truth is. Many philosophers con- 
^//der that the mind is naturally in possession of certain 
wh^ws or truths which it must recognise in every act of 
wh^iought ; all such, if they exist, would be a priori truths. 
dili cannot be doubted, for instance, that we must always 



XXIV.] AND SYNTHESi:^. ^09 

recognise in thought the three Primary Laws of Thought 
considered in Lesson XIV. We have there an a priori 
knowledge that "matter cannot both have weight and be 
without weight," or that "every thing must be either self- 
luminous or not self-luminous." But there is no law of 
thought which can oblige us co think that matter has 
weight, and luminous ether has not weight ; that Jupiter 
and Venus are not self-luminous, but that comets are to 
some extent self-luminous. These are facts which are no 
doubt necessary consequences of the laws of nature and 
the general constitution of the world ; but as we are not 
naturally acquainted with all the secrets of creation, we 
have to learn them by observation, or by' the a posteriori 
method. 

It is not however usual at the present time to restrict 
the name a priori to truths obtained altogether without 
recourse to observation. Knowledge may originally be 
of an a posteriori origin, and yet having been long 
in possession, and having acquired the greatest certainty, 
it may be the ground of deductions, and may then be said 
to give a priori knowledge. Thus it is now believed b^, 
all scientific men that force cannot be created or destroy 
ed by any of the processes of nature. If this be true th^ 
force which disappears when a bullet strikes a target mu,' 
be converted into something else, and on a priori gxonxn 
we may assert that heat will be the result. It is true th 
we might easily learn the same truth a posteriori, '; 
picking up portions of a bullet which has just struck 
target and observing that they are warm. But there is 
great advantage in a priori knowledge ; we can oft 
apply it in cases where experiment or observation wo^ 
be difficult. If I lift a stone and then drop it, the mc 
delicate instruments could hardly show that the stoi 
was heated by striking the earth ; yet on a priori gxour 
I know that it must have been so, and can easily caL 

14 



2IO PERFECT INDUCTION AND [less. 

late the amount of heat produced. Similarly we know, 
without the trouble of observation, that the Falls of Ni- 
agara and all other waterfalls produce heat. This is 
fairly an instance of a priori knowledge because no one 
that I have heard of has tried the fact or proved it a pos- 
teriori; nevertheless the knowledge is originally founded 
on the experiments of Mr Joule, who observed in certain 
well-chosen cases how much force is equivalent to a 
certain amount of heat. The reader, however, should 
take care not to con&ise the meaning of a priori thus 
explained with that given to the words by the philoso- 
phers who hold the mind to be in the possession of know- 
ledge independently of all observation. 

It is not difficult to see that the a priori method is 
equivalent to the synthetic method (see p. 205) considered 
in intension, the a posteriori method of course being equi- 
valent to the analytic method. But the same difference is 
really expressed in the words deductive and inductive; 
^ and we shall frequently need to consider it in the following 

P' lessons. 
in> 

For general remarks upon Method see the Port Royal 

pa ^ogic, Part iv. 

the 

be ) 

tha 

hett 

whii 

nat 



LESSON XXV. 

ERFECT INDUCTION AND THE INDUCTIVE 



opp 

nat SYLLOGISM. 

^ . E have in previous lessons considered deductive rea- 
'^^ ming, which consists in combining two or more general 
^ , :)positions synthetically, and thus arriving at a con- 
^ . ision which is a proposition or truth of less generality 



XXV.J THE INDUCTIVE SYLLOGISM. 211 

than the premises, that is to say, it applies to fewer indi- 
vidual instances than the separate premises from which 
it was inferred. When I combine the general truth that 
"metals are good conductors of heat," with the truth that 
'^aluminium is a metal," I am enabled by a syllogism in 
the mood Barbara to infer that "aluminium is a good con- 
ductor of heat." As this is a proposition concerning one 
metal only, it is evidently less general than the premise^ 
which referred to all metals whatsoever. In induction, on 
the contrary, we proceed from less general, or even from 
individual facts, to more general propositions, truths, or. 
as we shall often call them. Laws of Nature. When it is 
known that Mercury moves in an elliptic orbit round the 
Sun, as also Venus, the Earth, Mars, Jupiter, &c., we are 
able to arrive at the simple and general truth that "ail the 
planets move in elliptic orbits round the sun." This is an 
example of an inductive process of reasoning. 

It is true that we may reason without rendering our 
conclusion either more or less general than the premises, 
as in the following: — 

Snowdon is the highest mountain in England or Wales. 
Snowdon is not so high as Ben Nevis. 
Therefore the highest mountain in England or Wales i^ 
not so high as Ben Nevis. 

Again : 

Lithium is the lightest metal known. 

Lithium is the metal indicated by one bright red line :' 

the spectrum *. 
Therefore the lightest m.etal known is the metal indicat- 

by a spectum of one bright red line. 

In these examples all the propositions are singu' 
propositions, and merely assert the identity of singul 

* Roscoe's Lessons in Elejnentary Chemistry^ p. 199, 

14 — 2 



212 PERFECT INDUCTION AND Lless. 

terms, so that there is no alteration of generality. Each 
conclusion applies to just such an object as each of the 
premises applies to. To this kind of reasoning the apt 
name of Traduction has been given. 

Induction is a much more difficult and more important 
kind of reasoning process than Traduction or even Deduc- 
tion; for it is engaged in detecting the general laws or 
uniformities, the relations of cause and effect, or in short 
a-11 the general truths that may be asserted concerning the 
numberless and very diverse events that take place in the 
natural world around us. The greater part, if not, as 
some philosophers think, the whole of our knowledge, is 
ultimately due to inductive reasoning. The mind, it is 
plausibly said, is not furnished with knowledge in the 
form of general propositions ready made and stamped 
upon it, but is endowed with powers of observation, com- 
parison, and reasoning, which are adequate, when well 
educated and exercised, to procure knowledge of the world 
^ without us and the world within the human mind. Even 
P' when we argue synthetically and deductively from simple 
^^' ideas and truths which seem to be ready in the mind, as 
in the case of the science of geometry, it may be that we 
P^ lave gathered those simple ideas and truths from previous 
^^^)bservation or induction of an almost unconscious kind. 
•^^ '"his is a debated point upon which I will not here speak 
^^^ ositively ; but if the truth be as stated, Induction will be 
bett^^ mode by which all the materials of knowledge are 
^^-'rought to the mind and analysed. Deduction will then 
^^^,j the almost equally important process by which the 
^PP lowledge thus acquired is utilised, and by which nev/ 
nai .(iuctions of a more complicated character, as we shall 
^^y^, are rendered possible. 

^^^) An Induction, that is an act of Inductive reasonings is 
^^^ lied Perfect when all the possible cases or instances lo 

^^'^^cich the conclusion can refer, have been examined and 
dili ^ 



XXV.] THE INDUCTIVE SYLLOGISM. 213 

enumerated in the premises. If, as usually happens, it is 
impossible to examine all cases, since they may occur at 
future times or in distant parts of the earth or other 
regions of the universe, the Induction is called Imperfect. 
The assertion that all the months of the year are of less 
length than thirty-two days is derived from Perfect In- 
duction, and is a certain conclusion because the calendar 
is a human institution, so that we know beyond doubt how 
many months there are, and can readily ascertain that 
each of them is less than thirty-two days in length. But 
the assertion that all the planets move in one direction 
round the sun, from West to East, is derived from Imper- 
fect Induction ; for it is possible that there exist planets 
more distant than the most distant-known planet Nep- 
tune, and to such a planet of course the assertion would 
apply. \ 

Hence it is obvious that there is a great difference 
between Perfect and Imperfect Induction. The latter 
includes some process by which we are enabled to make 
assertions concerning things that we have never seen or 
examined or even known to exist. But it must be care- 
fully remembered also that no Imperfect Induction can 
give a certain conclusion. It may be highly probable or 
nearly certain that the cases unexamined will resemble 
those which have been examined, but it can never be 
certain. It is quite possible, for instance, that a new 
planet might go round the sun in an opposite direction to 
the other planets. In the case of the satellites belonging 
to the planets more than one exception of this kind has 
been discovered, and mistakes have constantly occurred 
in science from expecting that all new cases would 
exactly resemble old ones. Imperfect Induction thus 
gives only a certain degree of probability or likelihood 
that all instances will agree with those examined. Per- 
fect Induction, on the other hand, gives a necessary and 



214 PERFECT INDUCTION AND [less. 

certain conclusion, but it asserts nothing beyond what 
was asserted in the premises. 

Mr Mill, indeed, differs from almost all other logicians 
in holding that Perfect Induction is improperly called 
Induction, because it does not lead to any new knowledge. 
He defines Induction as inference from the kjiown to the 
unknown^ and considers the unexamined cases which are 
apparently brought into our knowledge as the only gaio 
from the process of reasoning. Hence Perfect Induction 
seems to him to be of no scientific value whatever, be- 
cause the conclusion is a mere reassertion in a briefer 
form, a mere summing up of the premises. I may point 
out, however, that if Perfect Induction were no more than 
a process of abbreviation it is yet of great importance, and 
requires to be continually used in science and common 
life. Without it we could never make a comprehensive 
statement, but should be obliged to enumerate every par- 
ticular. After examining the books in a library and 
finding them to be all English books we should be unable 
to sum up our results in the one proposition, " all the books 
in this library are English books ;" but should be required 
to go over the list of books every time we desired to make 
any one acquainted with the contents of the library. The 
fact is, that the power of expressing a great number of 
particular facts in a very brief space is essential to the pro- 
gress of science. Just as the whole science of arithmetic 
consists in nothing but a series of processes for abbreviat- 
ing addition and subtraction, and enabling us to deal with 
a great number of units in a very short time, so Perfect 
Induction is absolutely necessary to enable us to deal with 
a great number of particular facts in a very brief space. 

It is usual to represent Perfect Induction in the form 
of an Inductive Syllogism, as in the following instance :-— 
Mercury, Venus, the Earth, &c., all move round the sue 
from West to East. 



XXV.] THE INDUCTIVE SYLLOGISM, 215 

Mercury, Venus, the Earth, &c., are all the known Planets. 
Therefore all the known planets move round the sun from 
West to East. 

This argument is a true Perfect Induction because the 
conclusion only makes an assertion of all known planets, 
which excludes all reference to possible future discoveries ; 
and we may suppose that all the known planets have been 
enumerated in the premises. The form of the argument 
appears to be that of a syllogism in the third figure, 
namely Darapti^ the middle term consisting in the group 
of the known planets. In reality, however, it is not an 
ordinary syllogism. The minor premise states not that 
Mercury, Venus, the Earth, Neptune, &c., are contained 
among the known planets, but that they are those planets, 
or are identical with them. This premise is then a 
doubly universal proposition of a kind (p. 184 — 7) not re- 
cognised in the Aristotelian Syllogism. Accordingly we 
may observe that the conclusion is a universal proposi- 
tion, which is not allowable in the third figure of the syl- 
logism. 

As another example of a Perfect Induction we may 
take — ■ 

January, February, December, each contain less 

than 32 days. 

January December are all the months of the year. 

Therefore all the months of the year contain less than 32 

days. 

Although Sir W. Hamilton has entirely rejected the 
notion, it seems worthy of inquiry whether the Inductive 
Syllogism be not really of the Disjunctive form of Syllo- 
gism. Thus I should be inclined to represent the last 
example in the form :^ 

A month of the year is either January, or February, 
or March *..or December; but January has k»s 



21 6 PERFECT INDUCTION AND [less, 

than 32 days ; and February has less than 32 days ; aiid 
so on until we come to December, which has less than 
32 days. 

It follows clearly that a month must in any case have 
less than 32 days; for there are only 12 possible cases, 
and in each case this is affirmed. The fact is that the 
major premise of the syllogism on the last page is a 
compound sentence with twelve subjects, and is therefore 
equivalent to twelve distinct logical propositions. The 
minor premise is either a disjunctive proposition, as I have 
represented it, or something quite different from anything 
we have elsewhere had. 

From Perfect Induction we shall have to pass to Im- 
perfect Induction ; but the opinions of Logicians are not 
in agreement as to the grounds upon which we are war- 
ranted in taking a part of the instances only, and con- 
cluding that what is true of those is true of all. Thus if 
we adopt the example found in many books and say— 

This, that, and the other magnet attract iron ; 
This, that, and the other magnet are all magnets ; 
Therefore all magnets attract iron, 

we evidently employ a false minor premise, because this, 
that, and the other magnet which we have examined, 
cannot possibly be all existing magnets. In whatevei 
form we put it there must be an assumption that the mag- 
nets which we have examined are a fair specimen of all 
magnets, so that what we find in some we may expect in 
all. Archbishop Whately considers that this assumption 
should be expressed in one of the premises, and he repre- 
sents Induction as a Syllogism in Barbara as follows : — 

That which belongs to this, that, and the other magnet, 

belongs to all ; 
Attracting iron belongs to this, that, and the other; 
Therefore it belongs to all. 



XXV.] THE INDUCTIVE SYLLOGISM 217 

But though this is doubtless a correct expression of the 
assumption made in an Imperfect Induction, it does not 
in the least explain the grounds on which we are allowed 
to make the assumption, and under what circumstances 
such an assumption would be likely to prove true. Some 
writers have asserted that there is a Principle called the 
Uniformity of Nature, which enables us to affirm that 
what has often been found to be true of anything will 
continue to be found true of the same sort of thing. It 
must be observed, however, that if there be such a principle 
it is liable to exceptions; for many facts which have held 
true up to a certain point have afterwards been found not 
to be always true. Thus there was a wide and unbroken 
induction tending to show that all the Satellites m the 
planetary system went in one uniform direction round 
their planets. Nevertheless the Satellites of Uranus when 
discovered were found to move in a retrograde direction, 
or in an opposite direction to all Satellites previously 
known, and the same peculiarity attaches to the Satellite 
of Neptune more lately discovered. 

We may defer to the next lesson the question of the 
varying degree of certainty which belongs to induction in 
the several branches of knowledge. 

The advanced student may consult the following with 
advantage : — ManseFs Aldrich, Appendix, Notes G and H. 
Hamilton's Lectures on Logic ^ Lecture xvii., and Appen- 
dix VII., On Induction and Example^ Vol. II., p. 358. J. S. 
Mill's System of Logic ^ Book iii. Chap. 2, Of Inductions 
improperly so-called* 



2i8 INDUCTION, ANALOGY [LESS, 



LESSON XXVI. 

GEOMETRICAL AND MATHEMATICAL INDUC- 
TION, ANALOGY AND EXAMPLE. 

It is now indispensable that we should consider with 
great care upon what grounds Imperfect Induction is 
founded. No difficulty is encountered in Perfect Induc- 
tion because all possible cases which can come under the 
general conclusion are enumerated in the premises, so 
that in fact there is no information in the conclusion which 
was not given in the premises. In this respect the In- 
ductive Syllogism perfectly agrees with the general prin- 
ciples of deductive reasoning, which require that the in- 
formation contained in the conclusion should be shown 
only from the data, and that we should merely unfold, 
or transform into an explicit statement what is contained 
in the premises implicitly. 

In Imperfect Induction the process seems to be of a 
wholly different character, since the instances concerning 
which we acquire knowledge may be infinitely more 
numerous than those from which we acquire the know- 
ledge. Let us consider in the first place the process of 
Geometrical Reasoning which has a close resemblance to 
inductive reasoning. When in the fifth proposition of the 
first book of Euclid we prove that the angles at the base 
of an isosceles triangle are equal to each other, it is done 
by taking one particular triangle as an example. A 
figure is given which the reader is requested to regard as 
having two equal sides, and it is conclusively proved thaX 
if the sides be really equal then the angles opposite to 
ttiose sides must be equal also. But Euclid says nothing 
about other isosceles triangles ; he treats one single 
triangle as a sufficient specimen of all isosceles triangles, 



XXVL] AND EXAMPLE. 219 

and we are asked to believe that what is true of that is 
true of any other, whether its sides be so small as to be 
only visible in a microscope, or so large as to reach to the 
furthest fixed star. There may evidently be an infinite 
number of isosceles triangles as regards the length of the 
equal sides, and each of these may be infinitely varied by 
increasing or diminishing the contained angle, so that the 
number of possible isosceles triangles is infinitely infinite ; 
and yet we are asked to believe of this incomprehensible 
number of objects what we have proved only of one single 
specimen. This might seem to be the most extremely 
Imperfect Induction possible, and yet every one allows 
that it gives us really certain knowledge. We do know 
with as much certainty as knowledge can possess, that 
if lines be conceived as drawn from the earth to two stars 
equally distant, they will make equal angles with the line 
joining those stars ; and yet we can never have tried the 
experiment. 

The generality of this geometrical reasoning evidently 
depends upon the certainty with which we know that all 
isosceles triangles exactly resemble each other. The pro- 
position proved does not in fact apply to a triangle unless 
it agrees with our specimen in all the qualities essential 
to the proof. The absolute length of any of the sides or 
the absolute magnitude of the angle contained between 
any of them were not points upon which the proof de- 
pended — they were purely accidental circumstances ; 
hence we are at perfect liberty to apply to all new cases 
of an isosceles triangle what we learn of one case. Upon 
a similar ground rests all the vast body of certain know- 
ledge contained in the mathematical sciences — not only 
all t"he geometrical truths, but all general algebraical 
trutlb. It was shown, for instance, in p. 58, that if 
a and b be two quantities, and we multiply together 
their sum and difference, we get the difference of the 



aao INDUCTION, ANALOGY [less. 

squares of a and b. However often we try this it will be 
found true ; thus \i a — lo and b = y, the prpduct of the 
sum and difference is 17x3 = 51; the squares of the 
quantities are 10 x 10 or 100 and 7 x 7 or 49, the differ- 
ence of which is also 51. But however often we tried the 
rule no certainty would be added to it ; because when 
proved algebraically there was no condition which re^ 
stricted the result to any particular numbers, and a 
and b might consequently be any numbers whatever. 
This generality of algebraical reasoning by which a pro- 
perty is proved of infinite varieties of numbers at once, is 
one of the chief advantages of algebra over arithmetic. 
There is also in algebra a process called Mathematical 
Induction or Demonstrative Induction, which shows the 
powers of reasoning in a very conspicuous way. A good 
example is found in the following problem : — If we take 
the first two consecutive odd numbers, 1 and 3, and add 
them together the sum is 4, or exactly timce two; if we 
take three such numbers 1+34-5, the sum is 9 or exactly 
three times three; if we takey2?//r, namely 1 + 3 + 5 + 7 the 
sum is 16, or exactly y^^^** times four; or generally, if we 
take any given number of the series, 1+3 + 5 + 7 + .. . the 
sum is equal to the number of the terms multiplied by 
itself. Anyone who knows a very little algebra can prove 
that this remarkable law is universally true, as follows— 
Let n be the number of terms, and assume for a moment 
that this law is true up to n terms, thus — 

1+3 + 5 + 7 + •^'{^n—\) — n\ 

Now add 2/^ + 1 to each side of the equation. It fol- 
lows that — 

1 + 3 + 5 + 7 + + (2;?—i) + (2;2+i) = ;^^ + 2;?+i. 

But the last quantity ^^^ + 2« + i is just equal to {n + 1)'; 
so that if the law is true for n terms it is true also for ;/ -*- 1 
terms. We are enabled to argue from each single case of 



XXVI.] AND EXAMPLE, 221 

the law to the next case ; but we have already shown that 
it is true of the first few cases, therefore it must be true ot 
all. By no conceivable labour could a person ascertain by 
trial what is the sum of the first billion odd numbers, and 
yet symbolically or by general reasoning we know with 
certainty that they would amount to a billion billion, and 
neither more nor less even by a unit. This process of 
Mathematical Induction is not exactly the same as Geo- 
metrical Induction, because each case depends upon the 
last, but the proof rests upon an equally narrow basis of 
experience, and creates knowledge of equal certainty and 
generality. 

Such mathematical truths depend upon observation 
of a few cases, but they acquire certainty from the per- 
ception we have of the exact similarity of one case to 
another, so that we undoubtingly believe what is true of 
one case to be true of another. It is very instructive to 
contrast with these cases certain other ones where there 
is a like ground of observation, but not the same tie of 
similarity. It was at one time believed that if any integral 
number were multipled by itself, added to itself and then 
added to 41, the result would be a prime number, that is 
a number which could not be divided by any other in- 
tegral number except unity ; in symbols, 

x^ + x+41 = prime number. 

This was believed solely on the ground of trial and 
experience, and it certainly holds for a great many values 
of X. Thus when x is successively made equal to the 
numbers in the first line below, the expression x^ + x + ^i 
gives the values in the second line, and they are all prime 
numbers : 

0123456789 10 
41 43 47 53 61 71 83 97 113 131 151 

No reason however could be given why it should 



222 INDUCTION, ANALOGY [LESS. 

always be true, and accordingly it is found that the rule 
does not always hold true, but fails when x^^o. Then 
we have 40x40-1-40 + 41 = 1681, but this is clearly equal 
to 41 X 40 + 41 or 41 X 41, and is not a prime number. 

In that branch of mathematics which treats of the 
peculiar properties and kinds of numbers, other proposi- 
tions depending solely upon observation have been as- 
serted to be always true. Thus Fermat believed that 

2^+1 always represents a prime number, but could not 
give any reason for the assertion. It holds true in fact 
until the product reaches the large number 4294967297, 
which was found to be divisible by 641, so that the gene- 
rality of the statement was disproved. 

We find then that in some cases a single instance 
proves a general and certain rule, while in others a very 
great number of instances are insufficient to give any 
certainty at all ; all depends upon the perception we have 
of similarity or identity between one case and another. 
We can perceive no similarity between all prime numbers 
which assures us that because one is represented by a 
certain formula, also another is ; but we do find such 
similarity between the sums of odd numbers, or between 
isosceles triangles. 

Exactly similar considerations apply to inductions in 
physical science. When a chemist analyses a few grains 
of water and finds that they contain exactly 8 parts of 
oxygen and i of hydrogen for 9 parts of water, he feels 
warranted in asserting that the same is true of all pure 
w^ater whatever be its origin, and whatever be the part of 
the world from which it comes. But if he analyse a piece 
of granite, or a sample of sea- water from one part of the 
world, he does not feel any confidence that it will resem- 
ble exactly a piece of granite, or a sample of sea-water 
from another part of the earth ; hence he does not venture 
to assert of all granite or sea-water, what he finds true of 



XXVI.] AND EXAMPLE, 223 

a single sample. Extended experience shows that gra- 
nite is very variable in composition, but that sea-water is 
rendered pretty uniform by constant mixture of currents. 
Nothing but experience in these cases could inform us 
how far we may assert safely of one sample what v/e have 
ascertained of another. But we have reason to believe 
that chemical compounds are naturally fixed and invari- 
able in composition, according to Dalton's laws of com- 
bining proportions. No ^ priori reasoning from the 
principles of thought could have told us this, and we only 
learn it by extended experiment. But having once shown 
it to be true with certain substances we do not need to 
repeat the trial with all other substances, because we have 
every reason to believe that it is a natural law in which 
all chemical substances resemble each other. It is only 
necessary then for a single accurate analysis of a given 
fixed compound to be made in order to inform us of the 
composition of all other portions of the same substance. 

It must be carefully observed however that all induc- 
tions In physical science are only provable, or that if cer- 
tain, it is only hypothetical certainty they possess. Can 
I be absolutely certain that all water contains one part 
of hydrogen in nine ? I am certain only on two con- 
ditions : — 

1. That this was certainly the composition of the 
sample tried. 

2. That any other substance I call water exactly 
resembles that sample. 

But even if the first condition be undoubtedly true, I 
cannot be certain of the second. For how do I know 
what is water except by the fact of its being a transparent 
liquid, freezing into a SG^lid and evaporating into steam, 
possessing a high specific heat, and a number of other 
distinct properties ? But can I be absolutely certain that 
every liquid possessing all these properties is water? 



224 INDUCTION, ANALOGY [less. 

Practically I can be certain, but theoretically I cannot. 
Two substances may have been created so like each other 
that we should never yet have discovered the difference ; 
we might then be constantly misled by assuming of the 
one what is only true of the other. That this should ever 
happen with substances possessing the very distinct quali- 
ties of water is excessively imiprobable, but so far is it 
from being impossible or improbable in other cases, that 
it has often happened. Most of thS new elements dis- 
covered in late years have, without doubt, been mistaken 
previously for other elements. Caesium and Rubidium 
had been long mistaken for each other, and for Potassium, 
before they were distinguished by Bunsen and Kirchhoff 
by means of the spectroscope. As they are now known 
to be widely distributed, although in small quantities, it is 
certain that what was supposed to be Potassium in many 
thousands of analyses was partly composed of different 
substances. Selenium had probably been confused with 
Sulphur, and there are certain metals — for instance. Rho- 
dium, Ruthenium, Iridium, Osmium, and Beryllium- 
Yttrium, Erbium, Cerium, Lanthanum, and Didymium — 
Cadmium and Indium — which have only recently been 
distinguished. The progress of science will doubtless 
show that we are mistaken in many of our identifications, 
and various difficulties thus arising will ultimately be ex' 
plained. 

'Take again a very different case of induction. Are 
we certain that the sun will rise again to-morrow morning 
as it has risen for many thousand years, and probably for 
some hundred million years ? We are certain only on this 
condition or hypothesis, that the planetary system proceeds 
to-morrow as it has proceeded for so long. Many causes 
may exist which might at any moment defeat all our 
calculations ; our sun is believed to be a variable star, and 
for what we know it might at any moment suddenly 



XXVI.] AND EXAMPLE, 22% 

explode or flare up, as certain other stars have been ob- 
served to do, and we should then be all turned into thin 
luminous vapour in a moment of time. It is not at all 
impossible that a collision did once occur in the planet- 
ary system, and that the minute planets or asteroids are 
the result. Even if there is no large meteor, comet or 
other body capable of breaking up the earth by collision, 
yet it is probable that the sun moves through space at the 
rate of nearly 300 miles per minute, and if some other 
star should meet us at a similar rate the consequences 
would be inconceivably terrible. It is highly improbable 
however that such an event should come to pass even in 
the course of a million years. 

The reader will now see that no mere Imperfect In- 
duction can give certain knowledge ; all inference proceeds 
upon the assumption that new instances will exactly re- 
semble old ones in all material circumstances ; but in 
natural phenomena this is purely hypothetical, and we 
may constantly find ourselves in error. In Mathematical 
Induction certainty arose from the cases being hypotheti- 
cal in their own nature, or being made so as exactly to 
correspond with the conditions. We cannot assert that 
any triangle existing in nature has two equal sides or two 
equal angles, and it is even impossible in practice that 
any two lines or angles can be absolutely equal. But it 
is nevertheless true that if the sides are equal the angles 
are equal. All certainty of inference is thus relative and 
hypothetical. Even in the syllogism the certainty of the 
conclusion only rests on the hypothesis of certainty in the 
premises. It is probable, in fact, that all reasoning reduces 
itself to a single type — that what is true of one thing will 
be true of another thing, on condition of there being an 
exact resemblance between them in all material circum- 
stances. 

The reader will now understand with ease the nature 

15 



226 INDUCTION, ANALOGY [less, 

of reasoning by analogy. In strictness an analogy is not 
an identity of one thing with another, but an identity of 
relations. In the case of numbers 7 is not identical with 
10 nor 14 with 20, but the ratio of 7 to 10 is identical with 
the ratio of 14 to 20, so that there is an analogy between 
these numbers. To multiply two by two is not the same 
thing as to construct a square upon a line two units 
long ; but there is this analogy — that there will be just as 
many units of area in the square as there are units in the 
product of two by two. This analogy is so evident that 
we fearlessly assert a square mile to consist of 1760 x 1760 
square yards without any trial of the truth. In ordinary 
language, however, analogy has come to mean any re- 
semblance between things which enables us to believe of 
one what we know of the other. 

Thus the planet Mars possesses an atmosphere, with 
clouds and mist closely resembling our own ; it has seas 
distinguished from the land by a greenish colour, and 
polar regions covered with snow. The red colour of the 
planet seems to be due to the atmosphere, like the red 
colour of our sunrises and sunsets. So much is similar 
in the surface of Mars and the surface of the Earth 
that we readily argue there must be inhabitants there 
as here. All that we can certainly say however is, 
that if the circumstances be really similar, and similar 
germs of life have been created there as here, there must 
be inhabitants. The fact that many circumstances are 
similar increases the probability. But between the Earth 
and the Sun the analogy is of a much fainter character ; 
we speak indeed of the sun's atmosphere being subject to 
storms and filled with clouds, but these clouds are heated 
probably beyond the temperature of our hottest furnaces ; 
if they produce rain it must resemble a shower of melted 
iron ; and the sun-spots are perturbations of so tremend- 
ous a size and character, that the earth together with 



xxvi.j AND EXAMPLE. 22f 

half-a-dozen of the other planets could readily be swai* 
lowed up in one of them ■^. It is plain then that there is 
little or no analogy between the Sun and the Earth, and 
we can therefore with difficulty form a conception of any- 
thing going on in a sun or star. 

Argument from analogy may be defined as direct 
inductive inference from one instance to any similar 
instance. It may, as Mr Mill says, be reduced to the 
following formula : — 

"Two things resemble each other in one or more 
respects ; a certain proposition is true of the one ; there- 
fore it is true of the other." This is no doubt the type of 
all reasoning, and the certainty of the process depends 
entirely upon the degree of resemblance or identity be- 
tween the cases. In geometry the cases are absolutely 
identical in all material points by hypothesis, and no 
doubt attaches to the inference ; in physical science the 
identity is a question of probability, and the conclusion is 
in a like degree probable. It should be added that Mr 
Mill considers Geometrical and Mathematical Induction 
not to be properly called Induction, for reasons of which 
the force altogether escapes my apprehension ; but the 
reader will find his opinions in the 2nd chapter of the 
3rd book of his Syste^n of Logic, 

One form of analogical or inductive argument consists 
in the constant use of examples and instances. The best 
way to describe the nature of a class of things is to pie- 
sent one of the things itself, and point out the properties 
which belong to the class as distinguished from those 
peculiar to the thing. Throughout these Lessons, as 
throughout every work on Logic, instances of propositions, 
of compound or complex sentences, of syllogisms, &c., are 
continually used, and the reader is asked to apply to all 

* Lockyer's Elementary Lessons in Astronomy , § 108. 

15 — 2 



228 OBSEI^ VA TION [LESS. 

similar cases what he observes in the examples given. 
It is assumed that the writer selects such examples as 
truly exhibit the properties in question. 

While all inductive and analogical inferences rest 
»jpon the same principles there are wide differences be- 
tween the sources of probability. In analogy we have two 
cases which resemble each other in a great many proper- 
ties, and we infer that some additional property in one is 
probably to be found in the other. The very narrow 
basis of experience is compensated by the high degree of 
similarity. In the processes more commonly treated 
under the name Induction, the things usually resemble 
each other only in two or three properties, and we require 
to have more instances to assure us that what is true of 
of these is probably true of all similar instances. The 
less, in short, the intension of the resemblance the greater 
must be the extension of our inquiries. 

We proceed to the ordinary processes of Induction in 
the following Lessons. 

Mr MilFs System of Logic^ Book III. Chap. XX. Oj 
Analogy, Mansel's Aldrick, App. Note H. On 
Exafnple and Analogy, 



LESSON XXVIL 
OBSERVATION AND EXPERIMENT. 

All knowledge, it may be safely said, must be ultimately 
founded upon experience, which is but a general name for 
the various feelings impressed upon the mind at any period 
of its existence. The mind never creates entirely new 
knowledge independent of experience, and all that the 
reasoning powers can do is to arrive at the full meaning 



XXVII.] AND EXPERIMENT. 229 

of the facts which are in our possession. In previous 
centuries men of the highest abihty have held that the 
mind of its own power alone could, by sufficient cogita- 
tion, discover what things outside us should be, and 
would be found to be on examination. They thought 
that we were able to anticipate Nature by evolving 
from the human mind an idea of what things would be 
made by the Creator. The celebrated philosopher Des- 
cartes thus held that whatever the mind can clearly 
conceive may be considered true; but we can conceive 
the existence of mountains of gold or oceans of fresh 
water, which do not as a fact exist. Anything that we 
can clearly conceive must be conformable to the laws of 
thought, and its existence is then not impossible, so far as 
our intellect is concerned; but the forms and sizes and 
manners in which it has pleased the Creator to make 
things in this or any other part of the universe, cannot 
possibly be anticipated by the exceedingly limited wisdom 
of the human mind, and can only be learnt by actual ex- 
amination of existing things. 

In the latter part of the 13th century the great Roger 
Bacon clearly taught in England the supreme importance 
of experience as the basis of knowledge ; but the same 
doctrine was also, by a curious coincidence, again upheld 
in the 17th century by the great Chancellor Francis 
Bacon, after whom it has been called the Baconian PM- 
losophy. I believe that Roger Bacon was even a greater 
man than Francis, whose fame is best known ; but the 
words in which Francis Bacon proclaimed the importance 
of experience and experiment must be ever memorable. 
In the beginning of his great work, the Novum Organum, or 
New Instrument^ he thus points out our proper position 
as learners in the world of nature. 

"Man, the Servant and Interpreter of Nature, can do 
and understand as much as he has observed concerning 



230 OBSERVATION [less. 

the order of nature in outward things or in the mind; 
more, he can neither know nor do." 

The above is the first of the aphorisms or paragraphs 
with which the Novum Organum commences. In the 
second aphorism he asserts that the unaided mind can 
effect httle and is liable to err ; assistance in the form of 
a definite logical method is requisite, and this it was the 
purpose of his New Instrument to furnish. The 3rd and 
4th aphorisms must be given entire ; they are : — 

"Human science and human power coincide, because 
ignorance of a cause deprives us of the effect. For nature 
is not conquered except by obedience ; and what we dis- 
cover as a cause by contemplation becomes a rule in 
operation.'* 

"Man can himself do nothing else than move natural 
bodies to or from each other ; nature working within ac- 
complishes the rest." 

It would be impossible more clearly and completely 
to express the way in which we discover science by inter- 
preting the changes we observe in nature, and then turn 
our knowledge to a useful purpose in the promotion of 
the arts and manufactures. We cannot create and Ave 
cannot destroy a particle of matter ; it is now known that 
we cannot even create or destroy force ; nor can we really 
alter the inner nature of any substance that we have to 
deal with. All that we can do is to observe carefully how 
one substance by its natural powers acts upon another 
substance, and then by noving them together at the right 
time we can effect our object; as Bacon says, "Nature 
working within does the rest." Had it not been the 
nature of heat when applied to water to develope steam 
possessing elastic power, it is needless to say that the 
steam-engine could never have been made, so that the 
invention of the steam-engine arose from observing the 
utility of the force of steam, and employing it accordingly. 



XXVII.J AND EXPERIMENT, 231 

It is in this sense that Virgil has proclaimed him happy 
who knows the causes of things — 

Felix qui potuit rerum cognoscere causas^ 

and that Bacon has said, Knowledge is Power, So far 
as we have observed how things happen in nature, and on 
what occasion particu'lar effects are brought to pass, we 
are enabled to avoid or utilise those effects as we may 
desire, not by altering the natures of things, but by 
allowing them in suitable times and circumstances to 
manifest their own proper powers. It is thus, as Tenny- 
son has excellently said, that we 

" Rule by obeying Nature's Powers." 

Inductive logic treats of the methods of reasoning by 
which we may successfully interpret nature and learn the 
natural laws which various substances obey in different 
circumstances. In this lesson we consider the first requi- 
site of induction, namely, the experience or examination 
of nature which is requisite to furnish us with facts. Such 
experience is obtained either by observation or experiment. 
To observe is merely to notice events and changes which 
are produced in the ordinary course of nature, without 
being able, or at least attempting, to control or vary those 
changes. Thus the early astronomers observed the mo- 
tions of the sun, moon and planets among the fixed stars, 
and gradually detected many of the laws or periodical 
returns of those bodies. Thus it is that the meteorologist 
observes the ever-changing weather, and notes the height 
of the barometer, the temperature and moistness of the 
air, the direction and force of the wind, the height and 
character of the clouds, without being in the least able ta 
govern any of these facts. The geologist again is gene- 
nerally a simple observer when he investigates the nature 
and position of rocks. The zoologist, the botanist, ard 



?32 OBSER VA TION [less. 

the mineralogist usually employ mere observation when 
they examine animals, plants, and minerals, as they are 
met with in their natural condition. 

In experiment, on the contrar}'-, we vary at our will 
the combinations of things and circumstances, and then 
observe the result. It is thus that the chemist discovers 
the composition of water by using an electric current to 
separate its two constituents, oxygen and hydrogen. The 
mineralogist may employ experiment when he melts two 
or three substances together to ascertain how a particular 
mineral may have been produced. Even the botanist and 
zoologist are not confined to passive observation ; for by 
removing animals or plants to different climates and dif- 
ferent soils, and by what is called domestication, they 
may try how far the natural forms and species are capable 
of alteration. 

It is obvious that experiment is the most potent and 
direct mode of obtaining facts where it can be applied. 
We might have to wait years or centuries to meet acci- 
dentally with facts which we can readily produce at any 
moment in a laboratory ; and it is probable that most of 
the chemical substances now known, and many exces- 
sively useful products, would never have been discovered 
at all by waiting till nature presented them spontaneously 
to our observation. Many forces and changes too may 
go on in nature constantly, but in so slight a degree as to 
escape our senses, and render some experimental means 
necessary for their detection. Electricity doubtless ope- 
rates in every particle of matter, perhaps at every mo- 
ment of time ; and even. the ancients could not but notice 
its action in the loadstone, in lightning, in the Aurora 
Borealis, or in a piece of rubbed amber {electrum). But 
in lightning electricity was too intense and dangerous; 
in the other cases it was too feeble to be properly under- 
stood. The science of electricity and magnetism could 



I 



XXVII.] AND EXPERIMENT, 233 

only advance by getting regular, supplies of electricity 
from the common electric machine or the galvanic bat- 
tery, and by making powerful electro-magnets. Most if 
not all the effects which electricity produces must go on in 
nature, but altogether too obscurely for observation. 

Experiment, again, is rendered indispensable by the 
fact that on the surface of the earth we usually meet sub- 
stances under certain uniform conditions, so that we 
could never learn by observation what would be the 
nature of such substances under other conditions. Thus 
carbonic acid is only met in the form of a gas, proceeding 
from the combustion of carbon ; but when exposed to 
extreme pressure and cold, it is condensed into a liquid, 
and may even be converted into a snow-like solid sub- 
stance. Many other gases have in like manner been 
liquefied or solidified ; and there is reason to believe that 
every substance is capable of taking all the three forms of 
solid, liquid and gas, if only the conditions of temperature 
and pressure can be sufficiently varied. Mere observation 
of nature would have led us, on the contrary, to suppose 
that nearly all substances were fixed in one condition 
only, and could not be converted from solid into liquid 
and from liquid into gas. 

It must not be supposed however that we can draw 
any precise line between observation and experiment, and 
say where the one ends and the other begins. The dif- 
ference is rather one of degree than of kind ; and all we 
can say is that the more we vary the conditions artificially 
the more we employ experiment. I have said that me- 
teorology is a science of nearly pure observation, but if we 
purposely ascend mountains to observe the rarefaction 
and cooling of the atmosphere by elevation, or if we make 
balloon ascents for the same purpose, like Gay Lussac 
and Glaisher, we so vary the mode of observation as 
almost to render it experimental. Astronomers again 



234 OBSERVATION. [less. 

may almost be said to- experiment instead of merely OD- 
serving when they simultaneously employ instruments as 
far to the north, and as far to the south, upon the earth's 
surface as possible, in order to observe the apparent dif- 
ference of place of Venus when crossing the sun in a 
transit, so as thus to compare the distances of Venus and 
the sun with the dimensions of the earth. 

Sir John Herschel has excellently described the dif- 
ference in question in his Discourse on the Study of Na- 
tural Philosophy'^. " Essentially they are much alike, 
and differ rather in degree than in kind ; so that perhaps 
the terms passive and active observation might better 
express their distinction j but it is, nevertheless, highly 
important to mark the different states of mind in inqui- 
ries carried on by their respective aids, as well as their 
different effects in promoting the progress of science. 
In the former, we sit still and listen to a tale, told us, per- 
haps obscurely, piecemeal, and at long intervals of time, 
with our attention more or less awake. It is only by after 
rumination that we gather its full import ; and often, when 
the opportunity is gone by, we have to regret that our 
attention was not more particularly directed to some point 
which, at the time, appeared of little moment, but of 
which we at length appreciate the importance. In the 
latter, on the other hand, we cross-examine our witness, 
and by comparing one part of his evidence with the other, 
while he is yet before us, and reasoning upon it in his 
presence, are enabled to put pointed and searching ques- 
tions, the answer to which may at once enable us to make 
up our minds. Accordingly it has been found invariably, 
that in those departments of physics where the pheno- 
mena are beyond our control, or into which experimental 
enquiry, from other causes, has not been carried, the pro 

* p. 77. 



XXVII.] AND EXPERIMENT. 235 

gress of knowledge has been slow, uncertain and irregu- 
lar ; while in such as admit of experiment, and in which 
mankind have ag^reed to its adoption, it has been rapid, 
sure, and steady." 

Not uncommonly, however, nature has, so to speak, 
made experiments upon a scale and for a duration with 
which we cannot possibly compete. Thus we do not need 
to try the soil and situation which suits any given plant 
best ; we have but to look about and notice the habitat or 
situation in which it is naturally found in the most flou- 
rishing condition, and that, we may be sure, indicates the 
result of ages of natural experiment. The distances of 
the fixed stars would probably have been for ever un- 
known to us did not the earth by describing an orbit with 
a diameter of 1 82,000,000 miles make a sort of experimen- 
tal base for observation, so that we can see the stars in 
very slightly altered positions, and thus judge their dis- 
tances compared with the earth's orbit*. Eclipses, tran- 
sits, occultations and remarkable conjunctures of the pla- 
nets, are also kinds of natural experiments which have 
often been recorded in early times, and thus afford data 
of the utmost value. 

Logic can give little or no aid in making an acute or 
accurate observer. There are no definite rules which can 
be laid down upon the subject. To observe well is an art 
which can only be acquired by practice and training ; and 
it is one of the greatest advantages of the pursuit of the 
Natural Sciences that the faculty of clear and steady ob- 
servation is thereby cultivated. Logic can however give 
us this caution, which has been well pointed out by Mr 
Mill — to discriminate accurately between what we really 
do observe and what we only infer from the facts observed* 
So long as we only record and describe what our senses 

• See Lockyer's Elementary Lessons in Astronomy^ Nos. 
XLVI, XLVII. 



236 OBSERVATION [less 

have actually witnessed, we cannot commit an eiTor ; but 
the moment we presume or infer anything we are liable to 
mistake. For instance, we examine the sun's surface 
with a telescope and observe that it is intensely bright 
except where there are small breaks or circular openings 
in the surface with a dark interior. We are irresistibly 
led to the conclusion that the inside of the sun is colder 
and darker than the outside, and record as a fact that we 
saw the dark interior of the sun through certain openings 
in its luminous atmosphere. Such a record, however, 
would involve mistaken inference, for we saw nothing but 
dark spots, and we should not have done more in observ- 
ation than record the shape, size, appearance and change 
of such spots. Whether they are dark clouds above the 
luminous surface, glimpses of the dark interior, or, as is 
now almost certainly inferred, something entirely different 
froni either, can only be proved by a comparison of many 
unprejudiced observations. 

The reader cannot too often bear in mind the cau- 
tion against confusing facts observed with inferences from 
those facts. It is not too much to say that nine-tenths of 
what we seem to see and hear is inferred, not really felt. 
Every sense possesses what are called acquired percep- 
tions, that is, the power of judging unconsciously, by long 
experience, of many things which cannot be the objects of 
direct perception. The eye cannot see distance, yet we 
constantly imagine and say that we see things at such 
and such distances, unconscious that it is the result of 
judgment. As Mr Mill remarks, it is too much to say 
" I saw my brother." All I positively know is that J 
saw some one who closely resembled my brother as far 
as could be observed. It is by judgment only I can 
assert he was my brother, and that judgment may possi- 
bly be wrong. 

Nothing is more important in observation and experi- 



XXVII.] AND EXPERIMENT, 237 

ment than to be uninfluenced by any prejudice or theoi^ 
in correctly recording the facts observed and allowing to 
them their proper weight. He who does not do so will 
almost always be able to obtain facts in support of an 
opinion however erroneous. Thus the belief still exists 
with great force in the majority of uneducated persons, 
that the moon has great influence over the weather. The 
changes of the moon, full, new and half moon, occur four 
times in every month, and It is supposed that any change 
may influence the weather at least on the day preceding 
or following that of its occurrence. There will thus be 
twelve days out of every 28 on which any change of wea- 
ther would be attributed to the moon, so that during the 
J^ear many changes will probably be thus recorded as 
favourable to the opinion. The uneducated observer is 
struck with these instances and remembers them care- 
fully, but he fails to observe, or at least to remember, that 
changes of weather often occur also when there is no 
change of the moon at all. The question could only 
be decided by a long course of careful and unbiassed 
observation in which all facts favourable or unfavour- 
able should be equally recorded. All observations which 
have been published negative the idea that there can be 
any such influence as the vulgar mind attributes to the 
moon. 

But it would at the same time be an error to suppose 
that the best observer or experimentalist is he who holds 
no previous opinions or theories on the subject he inves- 
tigates. On the contrary, the great experimentalist is he 
who ever has a theory or even a crowd of theories or ideas 
upon his mind, but is always putting them to the test of 
experience and dismissing those which are false. The 
number of things which can be observed and experimented 
on are infinite, and if we merely set to work to record 
facts without any distinct purpose, our records will have 



238 OBSERVATION, &^c. [less 

no value. We must have some opmion or some the- 
or}^ to direct our choice of experiments, and it is more 
probable that we hit upon the truth in this way than 
merely by haphazard. But the great requisite of the 
true philosopher is that he be perfectly unbiassed and 
abandon every opinion as soon as facts inconsistent with 
it are observed. 

It has been well said by the celebrated Turgot, that 
" the first thing is to invent a system ; the second thing 
is to be disgusted with it;" that is to say, we ought to 
have some idea of the truth we seek, but should im- 
mediately put it to a severe trial as if we were inclined to 
distrust and dislike it rather than be biassed in its favour. 
Few men probably have entertained more false theories 
than Kepler and Faraday; few men have discovered or 
established truths of greater certainty and importance, 
Faraday has himself said that — 

" The world little knows how many of the thoughts 
and theories, which have passed through the mind of a 
scientific investigator, have been crushed in silence and 
secrecy by his own severe criticism and adverse examina- 
tion ; that in the most successful instances not a tenth of 
the suggestions, the hopes, the wishes, the preliminary 
conclusions have been realized*.'^ 

The student is strongly recommended to read Sir 
J, Herschel's Prelhninary Discourse on the Study 
of Natural Philosophy (Lardner's Cabinet Cyclo- 
pcedia), especially Part II. Chaps. 4 to 7, concerning 
Observation, Experiment, and the Inductive Pro 
cesses generally. 

* Modern Culture^ edited by Youmans, p. 222. [Macxnlllas 
And Co.] 



xxviji.] METHODS OF INDUCTION. 239 



LESSON XXVIII. 

METHODS OF INDUCTION. 

We have now to consider such methods as can be laid 
down for the purpose of guiding us in the search for gene- 
ral truths or laws of nature among the facts obtained by 
observation and experiment. Induction consists in infer- 
ring from particulars to generals, or detecting a general 
truth among its particular occurrences. But in physical 
science the truths to be discovered generally relate to 
the connection of cause and effect, and we usually call 
them laws of causation or natural laws. By the Cause of 
an event we mean the circumstances which must have 
preceded in order that the event should happen. Nor is 
it generally possible to say that an event has one single 
cause and no more. There are usually many different 
things, conditions or circumstances necessary to the pro- 
duction of an effect, and all of them must be considered 
causes or necessary parts of the cause. Thus the cause 
of the loud explosion in a gun is not simply the pulling of 
the trigger, which is only the last apparent cause or 
occasion of the explosion; the qualities of the powder; 
the proper form of the barrel ; the existence of some re- 
sisting charge ; the proper arrangement of the percussion 
cap and powder; the existence of a surrounding atmo- 
sphere, are among the circumstances necessary to the 
loud report of a gun : any of them being absent it would 
not have occurred. 

The cause of the boiling of water again is not merely 
the application of heat up to a certain degree of tempera* 



240 METHODS OF INDUCTION. [less. 

ture, but the possibility also of the escape of the vapour 
when it acquires a certain pressure. The freezing of 
water similarly does not depend merely upon the with- 
drawal of heat below the temperature of o° Centigrade. 
It is the work of Induction then to detect those circum- 
stances which uniformly will produce any given effect ; 
and as soon as these circumstances become known, we 
have a law or uniformity of nature of greater or less gene- 
rality. 

In this and the following Lessons I shall often have to 
use, in addition to cause and effect, the words antecedent 
and consequent, and the reader ought to notice their 
meanings. By an antecedent we mean any thing, condi- 
tion, or circumstance which exists before or, it may be, at 
the same time with an event or phenomenon. By a con- 
sequent we mean any thing, or circumstance, event, or 
phenomenon, which is different from any of the antecedents 
and follows after their conjunction or putting together. 
It does not follow that an antecedent is a cause, because 
the effect might have happened without it. Thus the 
sun's light may be an antecedent to the burning of a 
house, but not the cause, because the house would burn 
equally well in the night. A necessary or indispensable 
antecedent is however identical with a cause, being that 
without which the effect would not take place. 

The word phenomenon will also be often used. It 
means simply anything which appears, and is therefore 
observed by the senses ; the derivation of the word from 
the Greek word (paivojiepov, that which appears, exactly 
corresponds to its logical use. 

The first method of Induction is that which Mr Mill 
has aptly called the Metliod of agreement. It depends 
upon the rule that "If two or more instances of the phe- 
nomenon under investigation have only one circumstance 
in common, the circumstance in which alone all the in- 



XX nil.] METHODS OF INDUCTION, 241 

stances agree, is the cause (or effect) of the given pheno- 
menon." The meaning of this First Canon of inductive 
inquiry might, I think, be more briefly expressed by saying 
that the sole invariable antecedent of a phenomenon is 
probably its cause. 

To apply this method we must collect as many in- 
stances of the phenomenon as possible, and compare 
together their antecedents. Among these the causes will 
lie, but if we notice that certain antecedents are present or 
absent without appearing to affect the result, we conclude 
that they cannot be necessary antecedents. Hence it 
is the one antecedent or group of antecedents always 
present, when the effect follows, that we consider the cause. 
For example, bright prismatic colours are seen on bub- 
bles, on films of tar floating upon water, on thin plates 
of.mica, as also on cracks in glass, or between two pieces 
of glass pressed together. On examining all such cases 
they seem to agree in nothing but the presence of a very 
thin layer or plate, and it appears to make no appreciable 
difference of what kind of matter, solid, liquid, or gaseous, 
the plate is made. Hence we conclude that such colours 
are caused merely by the thinness of the plates, and this 
conclusion is proved true by the theory of the interference 
of light. Sir David Brewster beautifully proved in a 
similar way that the colours seen upon Mother-of-pearl 
are not caused by the nature of the substance, but by the 
form of the surface. He took impressions of the Mother- 
of-pearl in wax, and found that although the substance 
was entirely different the colours were exactly the same. 
And it was afterwards found that if a plate of metal had 
a surface marked by very fine close grooves, it would have 
iridescent colours like those of Mother-of-pearl. Hence 
it is evident that the form of the surface, which is the 
only invariable antecedent or condition requisite for the 
production of the colours, must be their cause 

16 



2A2 METHODS OF INDUCT/ON. [less. 

The method of agreement is subject to a serious 
difficulty, called by Mr Mill the Plurality of Causes, con- 
sisting in the fact that the same effect may in different 
instances be owing to different causes. Thus if we in- 
quire accurately into the cause of heat we find that it is 
produced by friction, by burning or combustion, by elec- 
tricity, by pressure, &c. ; so that it does not follow that if 
there happened to be one and the same thing present in 
all the cases we examined this would be the cause. The 
second method of induction which we will now consider 
is free from this difficulty, and is known as the Method of 
Difference. It is stated in Mr Mill's Second Canon as 
follows : — 

"If an instance in which the phenomenon under inves- 
tigation occurs, and an instance in which it does not 
occur, have every circumstance in common save one, that 
one occurring only in the former; the circumstance in 
which alone the two instances differ, is the effect, or the 
cause, or an indispensable part of the cause, of the phe- 
nomenon." 

In other words, we may say that the antecedent which 
is invariably present when the phenomenon follows, and 
invariably absent when it is absent, other circumstances 
remaining the same, is the cause of the phenomenon in 
those circumstances. 

Thus we can clearly prove that friction is one cause of 
heat, because when two sticks are rubbed together they 
become heated; when not rubbed they do not become 
heated. Sir Humphry Davy showed that even two pieces 
of ice when rubbed together in a vacuum produce heat, 
as shown by their melting, and thus completely demon- 
strated that the friction is the source and cause of the 
heat. We prove that air is the cause of sound being 
communicated to our ears by striking a bell in the re* 
ceiver of an air-pump, as Hawksbee first did in 1705, and 



XXVIII.] METHODS OF INDUCTION. an 

then observing that when the receiver is full of air we 
hear the bell ; when it contains little or no air we do 
not hear the bell. We learn that sodium or any of its 
compounds produces a spectrum having a bright yellow 
double line by noticing that there is no such line in the 
spectrum of light when sodium is not present, but that if 
the smallest quantity of sodium be thrown into the flame 
or other source of light, the bright yellow line instantly 
appears. Oxygen is the cause of respiration and life, 
because if an animal be put into a jar full of atmospheric 
air, from \/hich the oxygen has been withdrawn, it soon 
becomes suffocated. 

This is essentially the great metliod of experiment, 
and its utility mainly depends upon the precaution of only 
varying one circumstance at a tiine^ all other circum- 
sta7tces being maintained just as they were. This is 
expressed in one of the rules for conducting experiments 
given by Thomson and Tait in their great treatise on 
Natural Philosophy^ Vol. I. p. 307, as follows: — 

" In all cases when a particular agent or cause is to 
be studied, experiments should be arranged in such a way 
as to lead if possible to results depending on it alone ; or, 
if this cannot be done, they should be arranged so as to 
increase the effects due to the cause to be studied till 
these so far exceed the unavoidable concomitants, that 
the latter may be considered as only disturbing, not essen- 
tially modifying the effects of the principal agent." 

It would be an imperfect and unsatisfactory experi- 
ment to take air of which the oxygen has been converted 
into carbonic acid by the burning of carbon, and argue 
that, because an animal dies in such air, oxygen is the 
cause of respiration. Instead of merely withdrawing the 
oxygen we have a new substance, carbonic acid, present, 
which is quite capable of killing the animal by its own 
poisonous properties. The animal in fact would be suffo* 

16 — 2 



244 METHODS OF INDUCTION. [LESS, 

cated even when a considerable proportion of oxygen 
remained, so that the presence of the carbonic acid is a 
disturbing circumstance which confuses and vitiates the 
experiment. 

It is possible to prove the existence, and even to mea- 
sure the amount of the force of gravity, by delicately sus- 
pending a small ball about the size of a marble and then 
suddenly bringing a very heavy leaden ball weighing a 
ton or more close to it. The small ball will be attracted 
and set in motion; but the experiment would not be of the 
least value unless performed with the utmost precaution. 
It is obvious that the sudden motion of the large leaden 
ball would disturb the air, shake the room, cause currents 
in the air by its coldness or warmth, and even occasion 
electric attractions or repulsions; and these would pro- 
bably disturb the small ball far more than the force of 
gravitation. 

Beautiful instances of experiment according to this 
method are to be found, as Sir John Herschel has pointed 
out, in the researches by which Dr Wells discovered the 
cause of dew. If on a clear calm night a sheet or othei 
covering be stretched a foot or two above the earth, so 
as to screen the ground below from the open sky, dew will 
be found on the grass around the screen but not beneath 
it. As the temperature and moistness of the air, and other 
circumstances, are exactly the same, the open sky must 
be an indispensable antecedent to dew. The same expe- 
riment is indeed tried for us by nature, for if we make 
observations of dew during two nights which differ in no- 
thing but the absence of clouds in one and their presence 
in the other, we shall find that the clear open sky is requi- 
site to the formation of dew. 

It may often happen that we cannot apply the method 
of difference perfectly by varying only one circumstance 
at a time. Thus we cannot, generally speaking, try the 



XXVIII.] METHODS OF INDUCTION, 245 

• 

qualities of the same substance in the solid and liquid 
condition without any other change of circumstances, be- 
cause it is necessary to alter the temperature of the sub- 
stance in order to liquefy or solidify it. The temperature 
might thus be the cause of what we attribute to the liquid 
or solid condition. Under such circumstances we have 
to resort to what Mr Mill calls the joint method of agree- 
ment and difference, which consists in a double applica- 
tion of the method of agreement, first to a number of 
instances where an effect is produced, and secondly, to a 
number of quite different instances where the effect is not 
produced. It is clearly to be understood, however, that 
the negative instances differ in several circumstances 
from the positive ones ; for if they differed only in one 
circumstance we might apply the simple method of differ- 
ence. Iceland spar, for instance, has a curious power of 
rendering things seen through it apparently double. This 
phenomenon, called double refraction, also belongs to 
many other crystals ; and we might at once prove it to be 
due to crystalline structure could we obtain any transpa- 
rent substance crystallized and uncrystallized, but subject 
to no other alteration. We have, however, a pretty satis- 
factory proof by observing that uniform transparent un- 
crystallized substances agree in not possessing double 
refraction, and that crystalline substances, on the other 
hand, with certain exceptions which are easily explained, 
agree in possessing the power in question. The principle 
of the Joint method may be stated in the following rule, 
which is Mr MilFs Third Canon:— 

"If two or more instances in which the phenomenon 
occurs have only one circumstance in common, while two 
or more instances in which it does not occur have nothing 
in common save the absence of that circumstance ; the 
circumstance in which alone the two sets of instances 
(always or invariably) differ, is the effect, or the cause. 



2ifi METHODS OF INDUCTION. [less. 

or an indispensable part of the cause, of the pheno- 
menon.'* 

I have inserted the words in parentheses, as without 
them the canon seems to me to express exactly the oppo- 
site of what Mr Mill intends. 

It may facilitate the exact comprehension of these in- 
ductive methods if I give the following symbolic repre- 
sentation of them in the manner adopted by Mr Mill. 
Let A, By C, D, E^ &c., be antecedents which may be 
variously combined, and let «, b^ c, d^ e, &c., be effects 
following from them. If then we can collect the following 
sets of antecedents and effects — 



Antecedents. 


Consequents. 


ABC 


abc 


ADE 


ade 


AFG 


^fg 


AHK 


ahk 

> 



we may apply the method of agreement, and little doubt 

will remain that A^ the sole invariable antecedent, is the 

cause of a. 

The method of difference is sufficiently represented by — 

Antecedents. Consequents. 

ABC abc 

BC be 

Here while B and C remain perfectly unaltered we find 

that the presence or absence of A occasions the presence 

or absence of ^, of which it is therefore the cause, in the 

presence of B and C But the reader may be cautioned 

against thinking that this proves A to be the cause of a 

under all circumstances whatever. 

« 

The joint method of agreement and difference is similarly 

represented by — 



XXVIII.] METHODS OF INDUCTION. 247 



METHODS 


OF INDUCTION. 


Antecedents. 


Consequents. 


ABC 


abc 


ADE 


ade 


AFG 


^fg 


AHK 


ahk 


'Tq 


••• 

pq 


RS 


rs 


TV 


tv 


XY 


xy 



Here the presence of A is followed as in the simple method 
of agreement by a ; and the absence of -^, in circumstances 
differing from the previous ones, is followed by the ab- 
sence of a. Hence there is a very high probability that 
A is the cause of a. But it will easily be seen that A is 
not the only circumstance in which the two sets of in- 
stances differ, otherwise to any pair we might apply the 
simple method of difference. But the presence of ^ is a 
circumstance in which one set invariably, or uniformly, 
or always, differs, from the other set. This joint method is 
thus a substitute for the simpler method of difference in 
cases where that cannot be properly brought into action. 

HerschePs Discourse^ part II. chap. 6, p. 144. 
Mill's System of Logic ^ book ill. chaps. 8 and 9. 



LESSON XXIX. 

METHODS OF QUANTITATIVE INDUCTION. 

The methods of Induction described in the last Lesson 
related merely to the happening or not happening of the 
event, the cause of which was sought. Thus we learnt 
that friction was one cause of heat by observing that two 



t^S METHODS OF [less. 

solid bodies, even two pieces of ice, rubbed together, pro- 
duced heat, but that when they were not rubbed there 
was no such production of heat. This, however, is a very 
elementary sort of experiment ; and in the progress of an 
investigation we always require to measure the exact 
quantity of an effect, if it be capable of being more or 
less, and connecting it with the quantity of the cause. 
There is in fact a natural course of progress through 
which we proceed in every such inquiry, as may be stated 
in the following series of questions. 

1. Does the antecedent invariably produce an effect? 

2. In what direction is that effect.'^ 

3. How much is that effect in proportion to the cause? 

4. Is it uniformly in that proportion? 

5. If not, according to what law does it vary? 

Take for instance the effect of heat in altering the 
dimensions of bodies. The first question is, whether the 
heating of a solid body, say a bar of iron, alters its length ; 
the simple method of difference enables us to answer that 
it does. The next inquiry shows that almost all sub- 
stances are lengthened or increased in dimensions by 
heat, but that a very few, such as India rubber, and water 
below 4*o8° Cent., are decreased. We next ascertain the 
proportion of the change to each degree of temperature, 
which is called the coefficient of expansion. Thus iron 
expands 0*0000122 of its own length for every i® Centi- 
grade between o^ and looJJ. 

Still more minute inquiry shows, however, that the 
expansion is not uniformly proportional to temperature; 
most metals expand more and more rapidly the hotter 
they are, but the details of the subject need not be con- 
sidered here. 

The fixed stars, again, have often been mentioned in 
these Lessons, but the reader is probably aware that they 
are not really fixed. Taking any particular st^r, the 



XXIX.] QUANTITATIVE INDUCTION. 249 

astronomer has really to answer the several five questions 
stated below. 

Firstly. Does the star move ? 

2ndly. In what direction does it move? 

3rdly. How much does it move in a year or a century? 

4thly. Does it move uniformly? 

5thly. If not, according to what law does the motion 
vary in direction and rapidity? 

Every science and every question in science is first a 
matter of fact only, then a matter of quantity, and by 
degrees becomes more and more precisely quantitative. 
Thirty years ago most of the phenomena of electricity and 
electro-magnetism were known merely as facts ; now they 
can be for the most part exactly measured and calculated. 

As soon as phenomena can thus be measured we 
can apply a further Method of Induction of a very im- 
portant character. It is the Method of Difference indeed 
applied under far more favourable circumstances, where 
every degree and quantity of a phenomenon gives us 
a new experiment and proof of connection between cause 
and effect. It may be called the Method of Concomitant 
Variations, and is thus stated by Mr Mill, in what he 
entitles the Fifth Canon of Induction: 

*' Whatever phenomenon varies in any manner when- 
ever another phenomenon varies in some particular man- 
ner, is either a cause or an effect of that phenomenon, or 
is connected with it through some fact of causation." 

Sir John Herschel's statement of the same method is 
as follows : — " Increase or diminution of the effect, with the 
increased or diminished intensity of the cause, in cases 
which admit of increase and diminution," to which be 
adds, " Reversal of the effect with that of the cause." 

The illustrations of this method are infinitely nu- 
merous. Thus Mr Joule, of Manchester, conclusively 
proved that friction is a cause of heat by expending exaol 



2SO METHODS OF [less, 

quantities of force in rubbing one substance against 
another, and showed that the heat produced was exactly 
greater or less in proportion as the force was greater or 
less. We can apply the method to many cases which 
had previously been treated by the simple method of dif- 
ference; thus instead of striking a bell in a complete 
vacuum we can strike it with a very little air in the 
receiver of the air-pump, and we then hear a very faint 
sound, which increases or decreases every time we in- 
crease or decrease the density of the air. This experi- 
ment conclusively satisfies any person that air is the cause 
of the transmission of sound. 

It is this method which often enables us to detect the 
material connection which exists between two bodies. 
For a long time it had been doubtful whether the red 
flames seen in total eclipses of the sun belonged to the 
sun or the moon ; but during the last eclipse of the sun 
it was noticed that the flames inoved with the sun^ and 
were gradually covered and uncovered by the moon at 
successive instants of the eclipse. No one could doubt 
thenceforth that they belonged to the sun. 

Whenever, again, phenomena go through Periodic 
Changes, alternately increasing and decreasing, we should 
seek for other phenomena which go through changes in 
exactly the same periods, and there will probably be a 
connection of cause and effect. It is thus that the tides 
are proved to be due to the attraction of the moon and 
sun, -because the periods of high and low, spring and 
neap tides, succeed each other in intervals corresponding 
to the apparent revolutions of those bodies round the 
* earth. The fact that the moon revolves upon its own 
axis in exactly the same period that it revolves round the 
earth, so that for unknown ages past the same side of the 
moon has always been turned towards the earth, is a most 
perfect case of concomitant variations, conclusively prov- 



XXIX.] QUANTITATIVE INDUCTION. 251 

ing that the earth's attraction governs the motions of the 
moon on its own axis. 

The most extraordinary case of variations howevei 
consists in the connection which has of late years been 
shown to exist between the Aurora Borealis, magnetic 
storms, and the spots on the sun. It has only in the 
last 30 or 40 years become known that the magnetic 
compass needle is subject at intervals to very slight but 
curious movements ; and that at the same time there are 
usually natural currents of electricity produced in tele- 
graph-wires so as to interfere with the transmission of mes- 
sages. These disturbances are known as magnetic storms, 
and are often observed to occur when a fine display of 
the Northern or Southern Lights is taking place In some 
part of the earth. Observations during many years have 
shown that these storms come to their worst at the end of 
every eleven years, the maximum taking place about the 
present year 1870, and then diminish in intensity until 
the next period of eleven years has passed. Close obser- 
vations of the sun during 30 or 40 years have shown that 
the size and number of the dark spots, which are gigantic 
storms going on upon the sun's surface, increase and 
decrease exactly at the same periods of time as the mag- 
netic storms upon the earth's surface. No one can doubt, 
then, that these strange phenomena are connected to- 
gether, though the mode of the connection is quite un- 
known. It is now believed that the planets Jupiter, 
Saturn, Venus and Mars, are the real causes of the dis- 
turbances ; for Balfour Stewart and Warren de la Rue 
have shown that an exact correspondence exists between 
the motions of these planets and the periods of the sun- 
spots. This is a most remarkable and extensive case of 
concomitant variations. 

We have now to consider a method of Induction 
which must be employed when several causes act at once 



252 METHODS OF [less. 

and their effects are all blended together, producing a 
joint effect of the same kind as the separate effects. If 
in one experiment friction, combustion, compression and 
electric action are all going on at once, each of these 
causes will produce quantities of heat which will be added 
together, and it will be difficult or impossible to say how 
much is due to each cause separately. We may call this 
a case of the homogeneous intermixture of effects, the name 
indicating that the joint effect is of the same kind as 
the separate effects. It is distinguished by Mr Mill from 
cases of the heterogeneous, or, as he says, the hetero- 
pathic intermixture of effects, where the joint effect is 
totally different in kind from the separate effects. Thus 
if we bend a bow too much it breaks instead of bending 
further ; if we warm ice it soon ceases to rise in tempera- 
ture and melts ; if we warm water it rises in temperature 
homogeneously for a time but then suddenly ceases, and 
an effect of a totally different kind, the production of 
vapour, or possibly an explosion, follows. 

Now when the joint effect is of a heterogeneous kind 
the method of difference is sufficient to ascertam the cause 
of its occurrence. Whether a bow or a spring will break 
with a given weight may easily be tried, and whether 
water will boil at a given temperature in any given state 
of the barometer may also be easily ascertained. But ixv 
the homogeneous intermixture of effects we have a more 
complicated task. There are several causes each pro- 
ducing a part of the effect, and we want to know how 
much is due to each. In this case we must employ a 
further Inductive Method, called by Mr Mill the Methoa 
of Residues, and thus stated in his Fourth Canon : — 

" Subduct from any phenomenon such part as is known 
by previous inductions to be the effect of certain antece- 
dents, and the residue of the phenomenon is the effect of 
the remaining antecedents.'' 



xxrx.] QUANTITATIVE INDUCTION, 253 

If we know that the joint effect ^, b^ c is due to the 
causes A^ By and C, and can prove that a is due to A and 
b to By it follows that c must be due to C. There cannot 
be a simpler case of this than ascertaining the exact 
weight of any commodity in a cart by weighing the cart 
and load, and then subtracting the tare or weight of the 
cart alone, which had been previously ascertained. We 
can thiis too ascertain how much of the spring tides is 
due to the attraction of the sun, provided we have pre- 
viously determined the height of the tide due to the moon, 
which will be about the average height of the tides during 
the whole lunar month. Then subtracting the moon's 
tide the remainder is the sun's tide. 

Newton employed this method in a beautiful experi- 
ment to determine the elasticity of substances by allow- 
ing balls made of the substances to swing against each 
other, and then observing how far they rebounded com- 
pared with their original fall. But the loss of motion is 
due partly to imperfect elasticity and partly to the resist- 
ance of the air. He determined the amount of the latter 
effect in the simplest manner by allowing the balls to 
swing without striking each other, and observing how 
much each vibration was less than the last. In this way 
he was enabled easily to calculate the quantity that must 
be subtracted for the resistance of the air. 

It is this method that we employ in making allowance 
for the errors or necessary corrections in observations. 
Few thermometers are quite correct ; but if we put a ther- 
mometer into melting snow, which has exactly the tem- 
perature of o^ Centigrade, or 32^ Fahr., we can observe 
exactly how much below or above the true pomt the 
mercury stands, and this will indicate how much we 
ought to add or subtract from readings of the thermometer 
to make them correct. The height of the barometer is 
affected by several causes besides the variation of the 



254 METHODS OF lLESS. 

pressure of the air. It is decreased by the capillary 
repulsion between the glass tube and the mercury; it is 
increased by the expansion of the mercury by heat, if the 
temperature be above 32^ Fahr. ; and it may be increased 
or decreased by any error in the length of the measure 
employed to determine the height In an accurate obser- 
vation all these effects are calculated and allowed for in 
the final result. 

In chemical analysis this method is constantly em- 
ployed to determine the proportional weight of substances 
which combine together. Thus the composition of water 
is ascertained by taking a known weight of oxide of 
copper, passing hydrogen over it in a heated tube, and 
condensing the water produced in a tube containing sul- 
phuric acid. If we subtract the original weight of the 
condensing tube from its final weight we learn how much 
water is produced ; the quantity of oxygen in it is found 
by subtracting the final weight of the oxide of copper 
from its original weight. If we then subtract tke weight 
of the oxygen from that of the water we learn the weight 
of the hydrogen, which we have combined with the oxygen. 
When the experiment is very carefully performed, as de- 
scribed in Dr Roscoe's Lessons in Elementary Chemistry^ 
(p. 38), we find that 88'89 parts by weight of oxygen unite 
with 1 1 'I I parts of hydrogen to form 100 parts of water. 

In all sciences which allow of measurement of quan- 
tities this method is employed, but more especially in 
astronomy, the most exact of all the sciences. Almost all 
the causes and effects in astronomy have been found out 
as residual phenomena, that is, by calculating the effects of 
all known attractions upon a planet or satellite, and then 
observing how far it is from the place thus predicted. 
When this was very carefully done in the case of Uranus, 
it was still found that the planet was sometimes before 
and sometimes behind its true place. This residual effect 



xxix.J QUANTITATIVE INDUCTION, 255 

pointed to the existence of some cause of attraction not 
then known, but which was in consequence soon dis- 
covered in the shape of the planet Neptune. The motions 
of several comets have in this way been calculated, but it 
is observed that they return each time a little latet than 
they ought. This retardation points to the existence of 
nome obstructive power in the space passed through, the 
nature of which is not yet understood. 

Mill's Systein 0/ Logic, Book ill. Chap. 10, 0/ the 
Plurality of Causes j and of the Inter7nixture oj 
Effects, 



LESSON XXX. 

EMPIRICAL AND DEDUCTIVE METHODS. 

We have hitherto treated of Deduction and Induction as 
if they were entirely separate and independent methods. 
In reality they are frequently blended or employed alter- 
nately in the pursuit of truth ; and it may be said that all 
the more important and extensive investigations of science 
rely upon one as much as upon the other. It is probably 
the greatest merit in Mr MilFs logical writings that he 
points out the entire insufficiency of what is called the 
Baconian Method to detect the more obscure and difficult 
laws of nature. Bacon advised that we should always 
begin by collecting facts, classifying them according to 
their agreement and difference, and gradually gathering 
from them laws of greater and greater generality. He 
protested altogether against "anticipating nature," that is^ 
forming our own hypotheses and theories as to what the 
laws of nature probably are, and he seemed to think that 
systematic arrangement of facts would take the place 0/ 



256 EMPIRICAL AND DEDUCTIVE [LESS. 

all other methods. The reader will soon see that the 
progress of Science has not confirmed his opinions. 

When a law of nature is ascertained purely by induc- 
tion from certain observations or experiments, and has no 
other guarantee for its truth, it is said to be an empirical 
law. As Mr Mill says, "Scientific inquirers give the name 
of Empirical Laws to uniformities which observation or 
experiment has shown to exist, but on which they hesitate 
to rely in cases varying much from those which have been 
actually observed, for want of seeing any reason why 
such a law should exist." The name is derived from the 
Greek word €fX7T€if)La, meaning experience or trial. In- 
stances of such laws are abundant. We learn empiri- 
cally that a certain strong yellow colour at sunset, or an 
unusual clearness in the air, portends rain ; that a quick 
pulse indicates fever; that horned animals are always 
ruminants ; that quinine affects beneficially the nervous 
system and the health of the body generally ; that strych- 
nine has a terrible effect of the opposite nature : all these 
are known to be true by repeated observation, but we can 
give no other reason for their being true, that is, we 
cannot bring them into harmony with any other scientific 
facts ; nor could we at all have deduced them or antici- 
pated them on the ground of previous knowledge. The 
connection between the sun's spots, magnetic storms, 
auroras, and the motions of the planets mentioned in the 
last Lesson, is perhaps the most remarkable known 
instance of an empirical induction ; for no hint has yet 
been given of the way in which these magnetic influences 
are exerted throughout the vast dimensions of the planet- 
ary system. The qualities of the several alloys of metals 
are also good instances of empirical knowledge. No 
one can tell before mixing two or three metals for the 
first time in any given proportions what the qualities of 
the mixture will be — that brass should be both harder 



XXX.] METHODS, 257 

and more ductile than either of its constituents, copper 
and zinc ; that copper alloyed with the very soft metal tin 
should make hard and sonorous bell-metal ; that a certain 
mixture of lead, bismuth, tin and cadmium, should melt 
with a temperature (65® cent.) far below that of boiling 
water*. 

However useful may be empirical knowledge, it is yet 
of slight importance compared with the well-connected 
and perfectly explained body of knowledge which con- 
stitutes an advanced and deductive science. It is in 
fact in proportion as a science becomes deductive, and 
enables us to grasp more and more apparently uncon- 
nected facts under the same law, that it becomes perfect. 
He who knows exactly why a thing happens, will also 
know exactly in what cases it will happen, and what dif- 
ference in the circumstances will prevent the event from 
happening. Take for instance the simple effect of hot 
water in cracking glass. This is usually learnt empiri- 
cally. Most people have a confused idea that hot water 
has a natural and inevitable tendency to break glass, and 
that thin glass, being more fragile than other glass, will be 
more easily broken by hot water. Physical science, how- 
ever, gives a very clear reason for the effect, by showing 
that it is only one case of the general tendency of heat to 
expand substances. The crack is caused by the success- 
ful effort of the heated glass to expand in spite of the 
colder glass with which it is connected. But then we 
shall see at once that the same will not be true of thin 
glass vessels ; the heat will pass so quickly through that 
the glass will be nearly equally heated ; and accordingly 
chemists habitually use thin uniform glass vessels to hold 
or boil hot liquids without fear of the fractures which would 
be sure to take place in thick glass vessels or bottles. 

The history of science would show conclusively that 

* Roscoe's Lessons in Elementary Chemistry, p. 175. 

17 



258 EMPIRICAL AND DEDUCTIVE [less 

deduction was the clue to all the greatest discoveries. 
Newton, after Galileo the chief founder of experimen- 
tal philosophy, possessed beyond all question the great- 
est power of deductive thought which has ever been 
enjoyed by man. It is striking indeed to compare his 
results in optics with those in chemistry or alchemy. It 
is not generally known that Newton was really an alche- 
mist, and spent days and nights in constant experiments 
in his laboratory, trying to discover the secret by which 
metals could be transmuted into gold. But in these re- 
searches all was purely empirical, and he had no clue to 
guide him to successful experiments. A few happy 
guesses given in his celebrated Queries a^re all the result 
of this labour. But in the science of Optics it was quite 
otherwise ; here he grasped general laws, and every ex- 
periment only led him to devise and anticipate the results 
of several others, each more beautiful than the l^ast. Thus 
he was enabled to establish beyond all doubt the founda- 
tions of the science of the Spectrum, now bearing such 
wonderful results. Some persons may suppose that 
Newton, living shortly after Bacon, adopted the Baconian 
method, but I believe that there is no reference to Bacon 
in Newton's works ; and it is certain that he did not 
employ the method of Bacon. The Prineipia, though 
containing constant appeals to experiment and observa- 
tion, is nevertheless the result of a constant and sustained 
effort of deductive mathematical reasoning. 

What Mr Mill has called the Deductive Method, but 
which I think might be more appropriately called the 
Combined or Complete Method, consists in the alternate 
use of induction and deduction. It may be said to have 
three steps, as follows: — 

1. Direct Induction. 

2. Deduction, or, as Mr Mill calls it, Ratiocination. 

3. Verification. 



XXX. J METHODS. 259 

The first process consists in such a rough and simple 
appeal to experience as may give us a glimpse of the laws 
which operate, without being sufficient to establish their 
truth. Assuming them as provisionally true, we then 
proceed to argue to their effects in other cases, and a 
further appeal to experience either verifies or negatives 
the truth of the laws assumed. There are, in short, two 
appeals to experience connected by the intermediate use 
of reasoning. Newton, for instance, having passed a ray 
of sun-light through a glass prism found that it was spread 
out into a series of colours resembling those of the rainbow. 
He adopted the theory that white light was actually com- 
posed of a mixture of different coloured lights, which 
became separated in passing through the prism. He saw 
that if this were true, and he were to pass an isolated ray 
of the spectrum, for instance, the yellow ray, through a 
second prism, it ought not to be again broken up into 
different colours, but should remain yellow whatever was 
afterwards done with it. On trial he found this to be the 
case, and afterwards devised a succession of similar con- 
firmatory experiments which verified his theory beyond all 
possible doubt. 

It was no mere accident that led Pascal to have a 
barometer carried up to the top of the mountain Puy de 
Dome in France. Galileo, indeed, became acquainted by 
accident with the fact that water will not rise in an ordi- 
nary pump more than 33 feet, and was thus led to assert 
that the limited weight of the atmosphere caused it to 
rise. Torricelli, reasoning from this theory, saw that 
mercury, which is fourteen times as heavy as water, 
should not rise more than one -fourteenth part of the dis- 
tance, or about 29 or 30 inches. The experiment being 
tried verified the theory. It was the genius of Pascal, 
however, which saw that the experiment required to be 
varied in another way by carrying the mercurial barome- 

17 — 2 



26o EMPIRICAL AND DEDUCTIVE [LESa 

ter to the top of a mountain. If the weight of the atmo- 
sphere were really the cause of the suspension of the mer- 
cury, it ought to stand lowei on the mountain than below, 
because only the higher parts of the atmosphere pressed 
upon the mountain. The success of the experiment com- 
pletely verified the original hypothesis. The progress of 
the experimental sciences mainly depends upon the mode 
in which one experiment thus leads to others, and dis- 
closes new facts, which would in all probability have never 
come under our notice had we confined ourselves to the 
purely Baconian method of collecting the facts first and 
performing induction afterwards. 

The greatest result of the deductive method is no less 
than the theory of gravitation, which makes a perfect 
instance of its procedure. In this case the preliminary 
induction consisted, we may suppose, in the celebrated 
fall of the apple, which occurred while Newton was sitting 
in an orchard during his retirement from London, on 
account of the Great Plague. The fall of the apple, we 
are told, led Newton to reflect that there must be a power 
tending to draw bodies towards the earth, and he asked 
himself the question why the moon did not on that account 
fall upon the earth. The Lancashire astronomer Horrocks 
suggested to his mind another fact, namely, that when a 
stone is whirled round attached to a string, it exerts a 
force upon the string, often called centrifugal force. Hor- 
rocks remarked that the planets in revolving round the 
sun must tend in a similar way to fly ofl" from the centre. 
Newton was acquainted with Horrocks' views, and was 
thus possibly led to suppose that the earth's attractive 
force might exactly neutralise the moon's centrifugal 
tendency, so as to maintain that satellite in constant 
rotation. 

But it happened that the world was in possession of 
certain empirical laws concerning the motions of the pla- 



XXX.] METHODS, 261 

nets, without which Newton could scarcely have proceeded 
further. Kepler had passed a lifetime in observing the 
heavenly bodies, and forming hypotheses to explain their 
motions. In general his ideas were wild and unfounded, 
but the labours of a lifetime were rewarded in the esta- 
blishment of the three laws which bear his name, and 
describe the nature of the orbits traversed by the planets, 
and the relation between the size of such orbit and the 
time required by the planet to traverse it. Newton was 
able to show by geometrical reasoning that if one body 
revolved round another attracted Lo^.vard'^ it by a force 
decreasing as the square of the distance increases, it would 
necessarily describe an orbit of which Kepler's laws would 
be true, and which would tx*eref6re exactly resemble the 
orbits of the planets. Here was a partial verification of 
his theory by appeal to the results of experience. But 
several other philosophers had gone so far in the investi- 
gation of the subject. It is Newton's chief claim to ho- 
nour, that he carried on his deductions and verifications 
until he attained complete demonstration. To do this it 
was necessary first of all to show that the moon actually 
does fall towards the earth just as rapidly as a stone would 
if it were in the same circumstances. Using the best 
information then attainable as to the distance of the 
moon, Newton calculated that the moon falls through the 
space of 1 3 feet in one minute, but that a stone, if elevated 
so high, would fall through 15 feet. Most men would 
have considered this approach to coincidence as a proof 
of his theory, but Newton*s love of certain truth rendered 
him different even from most philosophers, and the dis- 
crepancy caused him to lay " aside at that time any fur- 
ther thoughts of this matter." 

It was not till many years afterwards (probably 15 
or 16) that Newton, hearing of some more exact data 
from which he could calculate the distance of the moon, 



262 EMPIRICAL AND DEDUCTIVE [less. 

was able to explain the discrepancy. His theoiy of gra- 
vitation was then verified so far as the moon was con- 
cerned ; but this was to him only the beginning of a long 
course of deductive calculations, each ending in a verifica- 
tion. If the earth and moon attract each other, and also 
the sun and the earth, similarly there is no reason why 
the sun and moon should not attract each other. Newton 
followed out the consequences of this inference, and showed 
that the moon would not move as if attracted by the 
earth only, but sometimes faster and sometimes slower. 
Comparisons with Flamstced's obi=^ervations of the moon 
showed that such was the case. Newton argued again, 
that as the waters of the ocean are not rigidly attached to 
the earth, they might attract the moon, and be attracted 
in return, independently of the rest of the earth. Certain 
daily motions would then be caused thereby exactly 
resembling the tides, and there were the tides to verify 
the fact. It was the almost superhuman power with 
which he traced out geometrically the consequences of his 
theory, and submitted them to repeated comparison with 
experience, which constitutes his preeminence over all 
philosophers. 

What he began has been going on ever since. The 
places of the moon and planets are calculated for each 
day on the assumption of the absolute truth of Newton's 
law of gravitation. Every night their places are observed 
as far as possible at Greenwich or some other observatory; 
comparison of the observed with the predicted place is 
always in some degree erroneous, and if coincident would 
be so only by accident. The theory is never proved com- 
pletely true, and never can be ; but the more accurately the 
results of the theory are calculated, and the more perfect 
the instruments of the astronomer are rendered, the more 
close is the correspondence. Thus the rude observations 
of Kepler and the few slight facts which worked on New- 



XXX.] METHODS. 263 

ton's mind, were the foundation of a theory which yielded 
indefinite means of anticipating new facts, and by con- 
stant verification, as far as human accuracy can go, has 
been placed beyond all reasonable doubt. 

Were space available it might be shown that all other 
great theories have followed nearly the same course. 
The undulatory theory of sound was in fact almost verified 
by Newton himself, though when he calculated from it 
the velocity of sound there was again a discrepancy, which 
only subsequent investigation could explain. This theory 
no doubt suggested the corresponding theory of light, 
which when followed out by Young, Fresnel, and others, 
always gave results which were ultimately in harmony 
with observation. It even enabled mathematicians to 
anticipate results which the most ardent imagination 
could hardly have guessed, and which mere haphazard 
experiment might never have revealed. Dalton's laws of 
equivalent proportions in chemistry, if not his atomic 
theory, were founded on experiments made with the 
simplest and rudest apparatus, but results deduced from 
them are daily verified in the nicest processes of modern 
chemical analysis. The still more modern theory of the 
Conservation of Energy, which had been vaguely antici- 
pated by Bacon, Rumford, Montgolfier, Seguin,* Mayer 
and possibly others, .was by Mr Joule brought to the test 
of experimental verification in some of the most beautiful 
and decisive experiments which are on record. It will be 
long before scientific men shall have traced out all the 
consequences of this grand principle, but its correspond- 
ence with fact already places it far beyond doubt. 

It will now be apparent, I think, that though observa- 
tion and induction must ever be the ground of all certain 
knowledge of nature, their unaided employment could 
never have led to the results of modern science. He who 
merely collects and digests facts will seldom acquire a 



264 EXPLANATION, TENDENCY, [less. 

comprehension of their laws. He who frames a theory 
and is content with his own deductions from it, like Des- 
cartes, will only surprise the world with his misused 
genius ; but the best student of science is he who with a 
copious store of theories and fancies has the highest 
power of foreseeing their consequences, the greatest dili- 
gence in comparing them, with undoubted facts, and the 
greatest candour in confessing the ninety-nine mistakes 
he has made in reaching the one true law of nature. 



LESSON XXXI. 

EXPLANATION, TENDENCY, HYPOTHESIS, 
THEORY, AND FACT. 

In the preceding Lessons I have used several expressions 
of which the meaning has not been defined. It will now 
be convenient to exempHfy the use of these terms, and tc 
arrive as far as possible at a clear understanding of their 
proper meanings. 

Explanation is literally the making plain or clear, so 
that there shall be nothing uneven or obscure to inter- 
rupt our view. Scientific explanation con?sists in harmo- 
nizing fact with fact, or fact with law, or law with law, 
so that we may see them both to be cases of one uniform 
law of causation. If we hear of a great earthquake m. 
some part of the world and subsequently hear that a 
neighbouring volcano has broken out, we say that the 
earthquake is thus partially explained. The eruption 
shows that there were great forces operating beneath the 
earth's surface, and the earthquake is obviously an effect 
of such causes. The scratches which may be plainly seen 
upon the surface of rocks in certain parts of Wales and 
Cumberland, are explained by the former existence of gla- 
ciers in those mountains; the scratches exactly harmonize 



XXXI.] HYPOTHESIS, THEORY, AND FACT. 265 

with the effects of glaciers now existing in Switzerland, 
Greenland, and elsewhere. These may be considered ex- 
planations of fact by fact. 

A fact may also be explained by a general law of 
nature, that is the cause and mode of its production may 
be pointed out and shown to be the same as operates in 
many apparently different cases. Thus the cracking of 
glass by heat was explained (p. 257) as one result of the 
universal law that heat increases the dimensions of solid 
bodies. The trade-winds are explained as one case of 
the general tendency of warm air to rise and be displaced 
by cold and dense air. The very same simple laws of heat 
and mechanics which cause a draught to flow up a chimney 
when there is a fire below, cause winds to blow from each 
hemisphere towards the equator. At the same time the 
easterly direction from which the winds come is explained 
by the simplest laws of motion ; for as the earth rotates 
from west to east, and moves much more rapidly at the 
equator than nearer the poles, the air tends to preserve 
its slower rate of motion, and the earth near the equator 
moving under it occasions an apparent motion of the wind 
from east to west. 

There are, according to Mr Mill, three distinct ways 
in which one law may be explained by other laws, or 
brought into harmony with them. 

The first is the case where there are really two 
or more separate causes in action, the results of which 
are combined or added together, homogeneously. As 
was before explained, homogeneous intermixture of effects 
(p. 252) means that the joint effect is simply the sum of the 
separate effects, and is of the same kind with them. Our 
last example of the trade-winds really comes under this 
case, for we find that there is one law or tendency which 
causes winds to blow from the arctic regions towards the 
equator, and a second tendency which causes then to blow 



266 EXPLANATION, TENDENCY, [LESS. 

from east to west. These tendencies are combined to- 
gether, and cause the trade-winds to blow from the North- 
East in the northern hemisphere, and from the South-East 
in the southern hemisphere. The law according to which 
the temperature of the air is governed in any part of the 
earth is a very complicated one, depending partly on the 
law by which the sun's heating power is governed, partly 
on the power of the earth to radiate the heat away into 
space, but even more perhaps on the effect of currents of 
air or water in bringing warmth or carrying i-t away. 
The path of a cannon-ball or other projectile is deter- 
mined by the joint action of several laws ; firstly, the 
simple law of motion, by which any moving body tends 
to move onward at an uniform rate in a straight line; 
secondly, the law of gravity, which continually deflects 
the body towards the earth's surface ; thirdly, the resist- 
ance of the air, which tends to diminish its velocity. 

The reader will perhaps have noticed the frequent use 
of the word tendency, and I have repeatedly spoken of a 
cause as tending to produce its effect. If the joint and 
homogeneous action of causes has been clearly explained, 
it will now be clear that a tendency means a cause which 
will produce an effect unless there be opposite causes, 
which, in combination wit'h it, counteract and disguise 
that effect. Thus when we throw a stone into the air the 
attractive power of the earth tends to make it fall, but the 
upward motion we have impressed upon it disguises the 
result for a certain time. The interminable revolving 
motion of the moon round the earth is the result of two 
balanced tendencies, that towards the earth, and that to 
proceed onward in a straight line. The laws of motion 
and gravity are such that this balance must always be 
preserved ; if the moon by any cause were brought nearer 
to the earth its tendency to fly off would be increased; 
and would exceed the effect of gravity until it had regained 



XXXI.] HYPOTHESIS, THEORY, AND FACT. 267 

its proper distance. A te7idency then is a cause which 
may or 7nay not be counteracted. 

In the second case of explanation an effect is shown 
to be due, not to the supposed cause directly, but to an 
intermediate effect of that cause. Instead oi A being the 
cause of C, it is found that A is the cause of B, and B the 
cause of C, so that B constitutes an intermediate link. 
This explanation may seem to increase the complexity of 
the matter, but it really simplifies it ; for the connection of 
A with B may be a case of a familiar and simple law, and 
so may that of B with C ; whereas the law that A pro- 
duces C may be purely empirical and apparently out of 
harmony with everything else. Thus in lightning it 
seems as if electricity had the power of creating a loud 
explosion ; but in reahty electricity only produces heat, 
and it is the heat which occasions sound by suddenly 
expanding the air. Thus thunder comes into harmony 
with the sound of artillery, which is also occasioned by 
the sudden expansion of the heated gases emitted by the 
powder. When chlorine was discovered it was soon found 
to have a strong power of bleaching, and at the present 
day almost all bleaching is done by chlorine instead of 
the sun, as formerly. Inquiry showed however that it was 
not really the chlorine which destroyed colour, but that 
oxygen is the intermediate and active agent. Chlorine 
decomposes water, and taking the hydrogen leaves the 
oxygen in a state of great activity and ready to destroy 
the organic colouring matter. Thus a number of facts 
are harmonized; we learn why dry chlorine does not 
bleach, and why there are several other substances which 
resemble chlorine in its bleaching power, for instance, 
ozone, peroxide of hydrogen, sulphurous acid, and .a pecu- 
liar oxide of vanadium, lately discovered by Dr Roscoe. 
It would be impossible to understand the effect at all un- 
less we knew that it is probably due to active oxygen or 



268 EXPLANATION, TENDENCY, [LESa 

ozone in all the cases, even in the old method of bleach- 
ing by exposure to the sun*. 

The third and much more important case of ex- 
planation is where one law is shown to be a ease of a 
more general law. As was explained in Lesson xxiv. we 
naturally discover the less general first, and gradually 
penetrate to the more simple but profound secrets of 
nature. It has often been found that scientific men were 
in possession of several well-known laws without perceiv- 
ing the bond which connected them together. Men, for 
instance, had long known that all heavy bodies tended to 
fall towards the earth, and before the time of Newton it 
was known to Hooke, Huyghens, and others, that some 
force probably connected the earth with the sun and moon. 
It was Newton, however, who clearly brought these and 
many other facts under one general law, so that each fact 
or less general law throws light upon every other. 

The science of Electricity now harmonizes a vast 
series of partial laws and facts between which it was 
a truly difficult task to discover any resemblance. The 
chief properties of the magnet had been fairly known 
since the time of Gilbert, the physician of Queen Eliza- 
beth ; common frictional electricity was carefully stu- 
died by Otto von Guericke, Epinus, Coulomb, and others ; 
Galvanism was elaborately investigated almost as soon 
as Galvani and Volta discovered the fact that the che- 
mical action of one substance on another may produce 
electricity. In the early part of this century there were 
three distinct sciences. Magnetism, Electricity and Gal- 
vanism ; now there is but one science. Oersted of 
Copenhagen gave in 1819 the first link between them, by 
pointing out that an electric current may cause move- 
ments in a compass-needle. Ampere and Faraday worked 

* Watts' Dictionary 0/ C/iemistryy Vol. I. p. 601. 



XXKI.] HYPOTHESIS, THEORY, AND FACT 269 

out the complicated relations of the three sciences, com- 
prehending them finally in a wider science, which may be 
called Electro-magnetism, or we may perhaps conveniently 
generalize the name Electricity so as to comprehend all 
the phenomena connected with it. 

A number of minor laws and detached facts are com- 
prehended and explained in the theory now generally 
accepted, that heat, electricity, light, and in fact all the 
phenomena of nature, are but manifestations in different 
forms of one same kind of energy. The total amount of 
energy existing in the universe is held to be fixed and un- 
alterable, like the quantity of matter ; sometimes it is 
disguised by affecting only the insensible molecules; at 
other times it is seen to produce palpable mechanical 
effects, as in the fall of a stone, or the expansion of 
steam. Now it had been previously known, ever since the 
time of the Greeks, that a simple lever, although greatly 
altering the character of force by making its action slower 
or faster, does not alter its amount, because the more 
intense the force the slower and more limited is its action. 
In modern times a similar truth was proved of every kind 
of machine; and it was recognised that, apart from friction, 
no kind of mechanism either creates or destroys energy. 
It had been independently recognised that electricity 
produced in the galvanic battery was exactly proportional 
to the amount of chemical action, and that almost any 
one of the forces named could be converted into any one 
of the others. All such facts are now comprehended 
under one general theory, the details of which are being 
gradually rendered more certain and accurate, but the 
main principle of which is that a certain amount of me- 
chanical energy is equal to a certain amount of heat, a 
certain amount of electricity, of chemical action, or even 
of muscular exertion. 

The word hypatliesis is much used in connection witlj 



270 EXP LAN A TION, TENDENCY, [less 

the subject we are discussing, and its meaning must be 
considered. It is derived from the Greek words xmoy 
under y and^eVty, //<^a;/^, and is. therefore exactly synony- 
mous with the Latin word supposition a placing under, 
whence our common word supposition. It appears to 
mean in science the imagining of some thing, force or 
cause, which underlies the phenomena we are examining, 
and is the agent in their production without being capable 
of direct observation. In making a hypothesis we assert 
the existence of a cause on the ground of the effects 
observed, and the probability of its existence depends 
upon the number of diverse facts or partial laws that we 
are thus enabled to explain or reduce to harmony. To be 
of any value at all a hypothesis must harmonize at least 
two different facts. If we account for the effects of opium 
by saying with Moliere that it possesses a dor77iitive 
power, or say that the magnet attracts because it has a 
magnetic power, every one can see that we gain nothing. 
We know neither more nor less about the dormitive or 
magnetic power than we do about opium or the magnet. 
But if we suppose the magnet to attract because it is 
occupied by circulating currents of electricity the hypo- 
thesis may seem a very improbable one, but is valid, 
because we thus draw a certain analogy between a magnet 
and a coil of wire conveying electricity. Such a coil of 
wire attracts other coils exactly in the way that one mag- 
net attracts another ; so that this hypothesis enables us 
to harmonize several different facts. The existence of 
intense heat in the interior of the earth is hypothetical in 
so far as regards the impossibility of actually seeing and 
measuring the heat directly, but it harmonizes so many 
facts derived from different sources that we can hardly 
doubt its existence. Thus the occurrence of hot springs 
and volcanoes are some facts in its favour, though they 
might be explained on other grounds ; the empirical law 



XXXI.] HYPOTHESIS, THEORY, AND FACT. 271 

that the heat increases as we sink mines in any part of 
the earth's surface is stronger evidence. The i-ntenselv 
heated condition of the sun and other stars is strongly 
confirmatory as showing that other bodies do exist in the 
supposed condition of the earth's interior. The cool 
state of the earth's surface is perfectly consistent with its 
comparatively small size and the known facts and laws 
concerning the conduction and radiation of heat. And 
the more we learn concerning the way in which the sun's 
heat is supplied by the fall of meteoric matter, the more 
it is probable that the earth may have been intensely 
heated like the sun at some former time, although for an 
immense period it has been growing slowly colder. A 
supposition coinciding with so many facts, laws, and other 
probable hypotheses, almost ceases to be hypothetical, 
and its high probability causes it to be regarded as a 
known fact. 

Provided it is consistent with the laws of thought there 
is nothing that we may not have to accept as a probable 
hypothesis, however difficult it may be to conceive and 
understand. The force of gravity is hypothetical in so 
far that we know it only by its effects upon the motions 
of bodies. Its decrease at a distance harmonizes exactly 
indeed with the way in which light, sound, electric or 
magnetic attractions, and in fact all influences which 
emanate from a point and spread through space, decrease ; 
hence it is probable that the law of the inverse square is 
absolutely true. But in other respects gravity is strongly 
opposed to all our ideas. If sound could travel to the 
sun as rapidly as in the earth's atmosphere it would re- 
quire nearly fourteen years to reach its destination ; were 
the sun and earth united by a solid continuous bar of iron, 
a strong pull at one end would not be felt at the other 
until nearly three years had passed. Light indeed comes 
from the sun in rather more than eight minutes ; but what 



272 EXPLANATION, TENDENCY, 

are we to think of the force of gravity, which appears tc 
reach the sun in an instant — so short that no calculations 
have yet been able to detect any interval at all ? In fact 
there seems some reason to suppose that gravity is felt 
instantaneously throughout the immeasurable regions of 
space. 

The undulatory hypothesis of light presents features 
equally extraordinary and inconceivable. That light does 
consist of minute but excessively rapid vibrations of 
something occupying space, is almost certain, because of 
the great harmony which this hypothesis introduces into 
the exceedingly various and complicated phenomena of 
light, and the explanation which it affords of the analogy 
of light to sound. It is difficult indeed to imagine that 
anything can oscillate so rapidly as to strike the retina 
of the eye 831,479,000,000,000 in one second, as must be 
the case with violet light according to this hypothesis. 
But this is nothing to the difiiculty of imagining space to 
be filled with solid ether of extreme rigidity and elasticity, 
but which nevertheless offers no appreciable resistance to 
the passage through it of ordinary matter, and does not 
itself possess any gravity *. It has been asserted indeed 
that the retardation in the return of comets is due to 
friction against this ether, and Mr Balfour Stewart be- 
lieves he has produced heat by friction of a metallic disc 
against the ether in a vacuum. Should these assertions 
prove to be true we have new facts in harmony with the 
theory of light, which would thereby become less hypo- 
thetical than before. 

There is no difficulty now in perceiving the part which 
hypothesis p'lays in the deductive method of scientific 
investigation considered in the iast lesson. The pre- 
hminary induction is replaced more or less completely b)f 

* See Sir John Herschel's Familiar Leetures, p. 315, &c 



xxxf.] HYPOTHESIS, THEORY, AND FACT, 273 

imagining the existence of agents which we think adequate 
to produce the known effects in question. If it is our 
object to explain the causes of ebbing and flowing wells, 
which occur in many parts of the world, we cannot 
possibly proceed by first exploring the interior of the 
earth, until we can discover the source of a spring, and 
observe its circumstances. We are obliged to imagine 
cavities and channels of various fonns, until we conceive 
such an apparatus as will, in accordance with known laws 
of hydrostatics^ occasion the irregular flowing of water in 
the way observed. If we can show that cavities of a 
particular form will produce that effect, and can think of 
no other mode in whieh it could be produced, the hypo- 
thesis becomes established as almost a certain fact. 

It is the same with any great hypothesis like that of the 
theory of light. We have no means of directly observing 
and measuring the qualities of the ether which is the 
medium of light. All we know about this ether at present 
is derived from the observed phenomena of light. Hence 
we are driven to invent something and endow it with 
qualities from which we may calculate, accordmg to some 
of the principles of mechanics, the effect to be expected ; 
and finding that these effects may be made to harmonize 
with those actually observed, we depend upon this coinci- 
dence to prove the existence of the ether. The truth of 
a hypothesis thus altogether depends upon subsequent 
verification and accordance with observed facts. To 
invent hypotheses which cannot thus be verified, or to 
invent them and then neglect the verification, leads to no 
result at all, or to fallacy. But when the verification is 
careful and complete no reproach can be brought against 
the employment of hypothesis. It becomes, perhaps, as 
certain as any other mode of investigation, and is at any 
rate indispensable. There was, in fact, little truth ox 
reason in Newton's celebrated protest against the use q\ 

i8 



274 EXPLANATION, TENDENCY, [less. 

hypothesis — " Hypothests non fingo." The fact is that as 
hi'j theory of gravitation rested upon the greatest and 
most successful of hypotheses, so his views of the material 
nature of light and the causes of its peculiar phenomena 
involved a false hypothesis, which has long since been 
completely disproved. 

The word theory has constantly been used in the 
last few lessons, and deserves some examination. It 
comes from the Greek Secopia, meaning contemplation, 
reflection or speculation; but this gives us little clue to its 
modern use. In reality the word is highly ambiguous, 
being sometimes used as equivalent to hypothesis, at 
other times as equivalent to general law or truth. When 
people form theories concerning comets, the sun, the 
cause of earthquakes, &c., they imagine a great many 
things which may or may not exist; such theories are 
really complicated hypotheses, and should be so called. 
In this sense there are two theories of electricity, one of 
which supposes the existence of a single fluid which 
accumulates in some places and has then a tendency to 
discharge itself towards places where there is a deficiency, 
just as water always tends to find its level ; the other 
supposes the existence of two fluids which are commonly 
united, but when separated tend to rush back into union 
again. These so-called theories are really hypotheses, be- 
cause we have no independent evidence of the existence 
of any fluid, and it is now almost certain that there is no 
such thing. The atomic theory, again, is really a hypo- 
thesis suggested by Dalton to explain the remarkable 
laws which he detected in the proportions of chemical 
elements which combine together. It is a valid hypothesis 
in so far as it does really explain the fixedness of the 
quantities which combine; but it is purely hypothetical 
as regards the shapes, properties or absolute magnitudes 
of the atoms, because we have no facts which it can har* 



XXXI ] HYPOTHESIS, THEORY, AND FACT, 275 

monise in these respects, and no apparent means of 
gaining them. 

In another and more proper sense theory is opposed 
to practice, just as the general is opposed to the particular. 
The theory of gravitation means all the more general laws 
of motion and attraction on which Newton founded his 
system of the Universe. We may know what those laws 
are without being able to determine the place of a planet 
or make any practical use of them ; the particular results 
must be calculated out by skilful astronomers before 
navigators, travellers or others can make practical use of 
them in the determination of the latitude or longitude. 
When we speak of the mathematical theory of sound, the 
lunar theory, the theory of the tides, the word is employed 
without any special reference to hypothesis, and is merely 
equivalent to general knowledge or science, implying the 
possession of a complete series of general and accurate 
laws, but in no way distinguishing them from accurate 
knowledge in general. When a word is really used in an 
equivocal manner like theory, it is not desirable to attempt 
to give it an accurate definition which would be imagi- 
nary and artificial. 

The word fact is used very often in this as in most 
books, and demands a few remarks. It is derived from 
factum, the past participle oi facere, to do, and would 
thus mean something which is done, an act, or deed ; but 
the meaning is evidently greatly extended by analogy. 
We usually oppose to each other fact and tlieory, but just 
as theory seems to have two ambiguous meanings, so 
I believe that fact is ambiguous. Sometimes it means 
what is certain and known by the evidence of the senses, 
as opposed to what is known only probably by hypothesis 
and inference; at other times it is contrasted to a general 
law, and is equivalent to a particular instance or case. A 
law oi great generality may often be as certain and true, 

18—2 



276 CLASSIFICATION, [less. 

especially in mathematics, as the particular facts coming 
under it, so that the contrast must in this case be that 
between the general and particular. We often use the 
word too in common life, as merely equivalent to truth; 
thus we might say, " It is a fact that the primary laws of 
thought are the foundation of reasoning.'' In short, as 
theory means ambiguously what is hypothetical, general, 
abstract or uncertain, so fact is equally ambiguous, and 
means confusedly what is intuitively known, particular, 
concrete or certain. 

Mill's System of Logic ^ Book in. Chapters 12, 13 and 
14, Of Explanation, and Hypothesis, 



LESSON XXXII. 

CLASSIFICATION, AND ABSTRACTION. 

In an earlier Lesson, upon the subject of the Predicables, 
we considered the doctrine of classification as it was 
treated by logicians many centuries ago. The progress 
of science, however, during the last two centuries has 
caused great attention to be given -to the true principles 
on which we can arrange a great multitude of diverse 
objects in order, and we have to consider what are the 
characteristics of a natural and perfect system of classifi- 
cation. 

It maybe said, indeed, that the subject we are treating 
is coextensive with the science of logic. All thought, all 
reasoning, so far as it deals with general names or general 
notions, may be said to consist in classification. Every 
common or general name is the name of a class, and every 
name of a class is a common name. " Metal" is the name 



XXXII.] AND ABSTRACTION. 277 

of one class of substances so often used in our syllogistic 
examples ; "Element" of another class, of which the former 
class is part. Reasoning has been plausibly represented 
to consist in affirming of the parts of a class whatever 
may be affirmed of the whole. Every law of nature which 
we arrive at enables us to classify together a number of 
facts, and it would hardly be too much to define logic as 
the theory of classification. 

Here we deal, however, with that more conscious and 
distinct arrangement of objects or notions, which is espe- 
cially employed in the natural sciences, such as Botany, 
Zoology, Mineralogy and Palaeontology. 

The derivation of the word class is somewhat curious. 
In ancient Rome it was the practice to summon the 
whole people together at certain periods, and this cere- 
mony was known as a clasis, from the Greek KXao-tf, or 
i^fjo-Ls, derived from /caXeco, to call together. Servius 
Tullius is said to have divided the people into six orders, 
according to the amount of tribute they could pay, and 
these orders were not unnaturally called the classes of the 
people. Hence the name came by degrees to be applied 
to any organized body of people, such as an army ; thence 
it was transferred to 'a fleet of vessels as marshalled in a 
fixed order, and was finally extended by analogy to any 
collection of objects carefully arranged. When, however, 
we now speak of the lower or^higher classes of the people 
it is curious that we are restoring the word very nearly -to 
its original'meaning. 

Classification may perhaps be best defined as the ar- 
rangement of things^ or our notions of the7n^ according to 
their resemblances or identities. Every class should so 
be constituted as to contain objects exactly resembling 
each other in certain definite qualities, which are stated 
XTi. the definition of the class. The more numerous and 
extensive the resemblances which are thus indicated by 



278 CLASSIFICATION, [les& 

any system of classes, the more perfect and useful must 
that system be considered. 

Mr Mill thus describes his view of the meaning— 
"Classification is a contrivance for the best possible 
ordering of the ideas of objects in our minds : for causing 
the ideas to accompany or succeed one another in such a 
way as shall give us the greatest command over our know* 
ledge already acquired, and lead mo^ directly to the 
acquisition of more. The general problem of classifica- 
tion, in reference to these purposes may be stated as 
follows : To provide that things shall be thought of in 
such groups, and those groups in such an order, as will 
best conduce to the remembrance, and to the ascertain- 
ment of their laws." 

A collection of objects may generally be classified in an 
indefinite number of ways. Any quality which is possess- 
ed by some and not by others may be taken as the first 
difference, and the groups thus distinguished may be sub- 
divided in succession by any other qualities taken at will. 
Thus a library of books might be arranged, (i) according 
to their size, (2) according to the language in which they 
are written, (3) according to the alphabetic order of their 
authors' names, (4) according to their subjects ; and in 
various other ways. lu large libraries and in catalogues 
such modes of arrangement are adopted and variously 
combined. Each different arrangement presents some 
peculiar convenience, and that mode must be selected 
which best meets the especial purpose of the library 
or catalogue. The population of a kingdom, again, may 
be classified in an almost endless number of ways with 
regard to different purposes or sciences. The popu- 
lation of the United Kingdom may be divided according 
to their place of birth, as English, Welsh, Scotch, Irisli, 
colonial-born, and aliens. The ethnographer would 
divide them into Anglo-Saxons, Cymri, Gaels, Picts, 



XXXII.] AND ABSTRACTION. 279 

Scandinavians, &c. The statist arranges them accord- 
ing to age ; to condition, as married, unmarried, widowed, 
&c. ; to state of body, as able, incapacitated, Wind, im- 
becile. The political economist regards the innumerable 
trades which are carried on, and classifies them in a 
complex manner. The lawyer again treats every one as a 
minor, an adult, a feme sole, a feme couverte, a guardian, 
ward, trustee, felon, and so on. 

In the natural world, again, we may make various 
classifications. Plants may be arranged according to the 
country from which they are derived ; the kind of place 
or habitat in which they flourish ; the time they live, as 
annual, biennial, perennial ; their size, as herbs, shrubs, 
trees; their properties, as esculents, drugs, or poisons: 
all these are distinct from the classifications which the 
botanist devises to represent the natural af^nities or 
relationships of plants. It is thus evident that in making 
a classification we have no one fixed method which can 
l»e ascertained by rule, but that an indefinite number of 
choices or alternatives are usually open to us. Logic 
cannot in such cases do much ; and it is really the work 
of the special sciences to investigate the character of the 
classification required. All that logic can do is to point 
out certain general requirements and principles. 

The first requisite of a good classification is, that it 
shall be appropriate to the purpose in hand ; that is to 
say, the points of resemblance selected to form the leading 
classes shall be those of importance to the practical use 
of the classification. All those things must be arranged 
together which require to be treated alike, and those 
things must be separated which require to be treated 
separately. Thus a lawyer has no need to classify per- 
sons according to the counties of England they were born 
in, because the law is the same independently cf counties ; 
but so far as a Scotchman, a Manx man, or an alien, is 



28o CLASSIFICATION, [less 

under diiferent laws from the English born man, we shall 
require to classify them apart. A gardener is quite right 
in classifying plants as annuals, biennials, perennials; as 
herbs, shrubs, trees; as evergreen and deciduous; or 
according to the soil, temperature and other circumstances 
which affect them, because these are points which must 
guide him in treating some differently from others. 

Another and, in a scientific point of view, the most 
important requisite of a good classification, is that it sliall 
enable tlie greatest possible mimber of general assertions 
to be made. This is the criterion, as stated by Dr 
Whewell, which distinguishes a natural from an artificial 
iystem of classification, and we must carefully dwell upon 
its meaning. It will be apparent that a good classification 
is more than a mere orderly arrangement ; it involves a 
process of induction which will bring to light all the more 
general relations which exist between the things classified. 
An arrangement of books will generally be artificial ; the 
octavo volumes will not have any common character ex- 
cept being of an octavo size. An alphabetical arrange- 
ment of names again is exceedingly appropriate and con- 
venient to many purposes, but is artificial because it 
allows of few or no general assertions. We cannot make 
any general assertion whatever about persons because 
their names happen to begin with an A or a B, a P or a 
W. Even those who agree in bearing the name Smith or 
Taylor or Robinson might be submitted to the inductive 
method of agreement without the discovery of any 
common circumstance which could be stated in a general 
proposition or law. It is true that if we investigated the 
antecedents of the Evanses and Joneses we should find 
them nearly all to be Welsh, and the Campbells to be 
Scotch, and those who bear a very peculiar name would 
often be found to descend from common ancestors. Sc 
tar even an alphabetic arrangement embodies some-thing 



xxxil.] AND ABSTRACTION. 281 

that is natural in it, and enables general assertions to be 
made. Hardly any arrangement can be made, in fact, 
which will not indicate some vestiges of important rela- 
tions and resemblances ; but what we want is a system 
which will reveal all the most important general truths. 

For this purpose we must select as the ground of 
union those characters which carry with them most other 
characters. In Lesson xii. we considered the proprium' 
as a quality which belongs to the whole of a class without 
forming part of the definition of the class. Now we 
ought to frame the definition of a class that it may con- 
tain as few characters as possible, but that as many other 
characters, properties, or propria^ as possible, shall be 
attributable to the things contained in the class. Every 
one can see, for instance, that animals form one great 
group of beings, which have many characters in common, 
and that plants form another group. Animals have sen- 
sation, voluntary motion, consume carbonaceous food, and 
evolve carbonic acid, possess a stomach, and produce 
fat. Plants are devoid of sensation and voluntary motion, 
produce carbonaceous tissue, absorb carbonic acid, and 
evolve oxygen, possess no stomach, and produce starch. 
At one time it might have been thought that almost any 
of the characters named was a sufficient mark of the 
group to which a being belonged. Whatever had a 
stomach, was an animal ; whatever had not, was a plant ; 
whatever produced starch or evolved oxygen was called a 
plant ; whatever absorbed oxygen or produced fat was an 
animal. To the present day these statements remain 
generally true, so that we may make assertions in the form 
of the proposition U, that " all animals are all beings 
that evolve carbonic acid, and all plants are all beings 
that absorb carbonic acid." But in reality the exceptions 
are many, and increasing research makes it continually 
more apparent that there is no definite line to be drawn 



282 CLA SSIFICA TlOn, [LESS. 

between animal and vegetable life. This, of course, is 
not a failure of logical science, but a fact of great sig- 
nificance concerning the things themselves. 

In a classification of plants we meet again with most 
deep and natural distinctions between the great classes 
called Exogens, Endogens, and Acrogens. The latter 
have no true sexual flowers and seeds, are formed almost 
whoUv of cellular tissue, and have an epidermis without 
cuticular pores. The former two classes have much in 
common ; they have true flowers, woody tissue and 
cuticular pores, and hence may be united into one wider 
class, Vasculares. But exogens and endogens are also 
most strongly distinguished. Exogens have a stem or 
trunk consisting of distinct bark, pith, and wood in con- 
centric layers, leaves with reticular veins, seeds with two 
seed-leaves and a naked radicle ; generally speaking, too, 
the parts of the flower are some multiple of two or five in 
number. Endogens, on the contrary, have no distinct 
bark, pith, and wood, no concentric layers, leaves with 
parallel veins, seeds with one seed-leaf, and a radicle not 
naked ; they have, too, the parts of the flower generally a 
multiple of three in number. 

These are the very widest classes in what is called 
the natural system of botanical arrangement ; but similar 
principles are observed in all its minor classes. The 
continual efforts of botanists are directed to bringing the 
great multitudes of plants together in species, genera, 
orders, classes, and in various intermediate groups, so 
that the members of each group shall have the greatest 
number of points of mutual resernblance and the fewest 
points of resemblance to members of other groups. Thus 
is best fulfilled the great purpose of classification, which 
reduces multiplicity to unity, and enables us to infer of al3 
tlie other members of a class what we know of any ona 
member, provided we distinguish properly between those 



XXXTL] AND ABSTRACTION. 283 

qualities which are hkely or are known to belong to the 
class, and those which are peculiar to the individual. It 
is a necessary condition of correct classification, as re- 
marked by Prof. Huxley, that the definition of a group 
shall hold exactly true of all members of the group, and 
not of the members of any other group. To carry out this 
condition in the natural sciences is, however, very difficult, 
because kinds of plants or animals are continually dis- 
covered which stand in an intermediate position between 
classes which would otherwise be well distinguished. 
Thus ferns much embarrass the fundamental division of 
plants, because though they have no true flowers, and in 
this and other respects agree with other acrogens, yet 
they have abundance of woody fibre, which would entitle 
them to rank with vasculares, the larger group of which 
exogens and endogens are the subdivisions. 

It may be remarked that the progress of chemistry is 
rapidly rendering it a science of classification ; and in fact 
the whole theory of chemical combination now depends 
on a correct grouping of elements and compounds. Dr 
Roscoe in his Lessons iji Elementary Chemistry enu- 
merates no less than eleven classes of metals, each class 
having a number of properties in common. Thus the 
metals of the alkalies, namely, Potassium, Sodium, Caesium, 
Rubidium, Lithium, form a remarkably natural class. 
They are all soft, easily fusible, volatile at high tempera- 
tures ; they combine with great force with oxygen, decom- 
pose water at all temperatures, forming oxides which are 
very soluble in water, and become powerfully caustic and 
alkaline bodies from which water cannot be expelled by 
heat. Their carbonates are soluble in water, and each 
metal forms only one compound with chlorine. 

The metals of the alkaline earths, Calcium, Strontium, 
and Barium, also form a very natural class, distinguished 
by the fact that their carbonates are insoluble in pure 



3^4 CLASSIFICATION, [les& 

water, but soluble in water containing carbonic acid in 
solution. The gold class contains the rare or valuable 
metals Gold, Platinum, Palladium, Rhodium, Ruthenium, 
Iridium, and Osmium, which are not acted on by nitric 
acid, and can only be dissolved by chlorine or the mixture 
of acids called aqua regia. The oxides can be reduced 
or deoxidised by simply heating them. 

Natural classifications give us the deepest resemblances 
and relations, and may lead us ultimately to a knowledge 
of the way in which the varieties of things are produced. 
They are, therefore, essential to a true science, and may 
almost be said to constitute the framework of the science. 
Yet it does not follow that they are appropriate for all 
purposes. When our purpose is merely to recognise the 
name of a chemical element, a plant or an animal, its 
character as defined in a natural system would give us 
little or, no assistance. The chemist does not detect 
potassium by getting it into the state of metal, and trying 
whether it would decompose water. He merely observes 
which, among all the compounds of potassium, have the 
best marked and most peculiar characters ; thus a com- 
pound of potassium, platinum, and chlorine is most 
distinctive or characteristic of the metal, and is generally 
used as a means of recognising it ; but a fine violet 
colour which potash gives to the flame of a lamp was 
also used as an indication of its presence long before 
the spectroscope was introduced to analyse such colours. 
An artificial classification of the elements is thus ne- 
cessary to the detection of substances, and accordingly 
in any book on chemical analysis will be found arrange- 
ments of the elements according to characters of very 
minor importance, but which are selected on arcount of 
the ease and certainty with which they can be observed. 

In Botany, again, the natural system of classification is 
fiar from being well suited for determining the name of a 



XXXII.] AND ABSTRACTION. 285 

plant, because the classes are often defined by the form of 
minute parts of the seed, the arrangement of the seed- 
vessel, and other parts which it is usually difficult or 
sometimes impossible to examine. Accordingly botanists 
usually arrange their genera and species in the order of 
the natural system, but contrive a sort of key or artificial 
arrangement, in which the most simple and apparent 
characters, often called characteristics, are employed for 
the discrimination of the plants. The best arrangement 
of this kind as regards British plants is to be found in 
Bentham's British Flora. In reality the celebrated 
Linnsean arrangement of plants was intended by its 
author to serve in this way. Linnaeus was too profound 
a philosopher to suppose that the numbers of stamens 
and pistils usually expressed the real relationships of 
plants. Many of his classes were really natural classes, 
but the stamens and pistils were selected as the general 
guide to the classes and orders, as being very plain and 
evident marks. 

Closely connected with the process of classification 
is that of abstraction. To abstract is to separate the 
qualities common to all individuals of a group from the 
peculiarities of each individual. The notion " triangle " 
is the result of abstraction in so far as we can reason 
concerning triangles, without any regard to the particular 
size or shape of any one triangle. All classification im- 
plies abstraction, for in framing and defining the class 
I must separate the common qualities from the peculiari- 
ties. When I abstract, too, I form a general conception, 
or one which, generally speaking, embraces many objects. 
If, indeed, the quality abstracted is a peculiar property of 
the class, or one which belongs to the whole and not to 
any other objects, I may not increase the extent of the 
notion, so that Mr Herbert Spencer is, perhaps, right in 
holding that we can abstract without generalizing. We 



286 CLASSIFICATION, &c, [less. 

often use this word generalization, and the process may be 
defined as inferring of a whole class what we know only oi 
a part. Whenever we regard the qualities of a thing as 
not confined to that thing only but as extended to other 
objects ; when, in fact, we consider a thing only as a 
member of a class, we are said to generalize. If, after 
studying the properties of the circle, we proceed to those 
of the ellipse, parabola and hyperbola, it is soon found 
that the circle is only one case of a whole class of curves 
called the conic sections, corresponding to equations of 
the second degree; and I generalize when I regard cer- 
tain of the properties of the circle as shared by many 
other curves. 

Dr Whewell added to the superabundance of terms to 
express the same processes when he introduced the ex- 
pression Colligation of facts. Whenever two things are 
found to have similar properties so as to be placed in the 
same class they may be said to be connected together. 
We connect together the places of a planet as it moves 
round the sun, when we conceive them as points upon a 
common ellipse. Whenever we thus join together pre- 
viously disconnected facts, by a suitable general notion or 
hypothesis, we are said to colligate them. Dr Whewell 
adds that the general* conceptions employed must be 
(i) clear, and (2) appropriate ; but it may well be ques- 
tioned whether there is anything really different in these 
processes from the general process of natural classification 
which we have considered. 



xxxiii.] OF A PHILOSOPHICAL LANGUAGE. 287 



LESSON XXXIII. 

REQUISITES OF A PHILOSOPHICAL 
LANGUAGE. 

Among the subsidiary processes requisite to the successful 
prosecution of inductive reasoning must be placed the 
construction of a suitable language. It is in fact impos- 
sible to over-estimate the importance of an accurate and 
copious language in any science ; and the study of things 
would be almost useless without names to denote those 
things and record our observations concerning them. 

It is easily apparent, indeed, that language serves 
three distinct and almost independent purposes : — 

1. As a means of communication. 

2. As a mechanical aid to thought. 

3. As an instrument of record and reference. 

In its first origin language was used chiefly if not exclu- 
sively for the first purpose. Savage tribes exist in great 
numbers at the present day who seem to accumulate no 
knowledge. We may even say that the lower animals 
often possess some means of communication by sounds 
or natural signs which constitute language in the first 
sense, though they are incapable of reasoning by general 
notions. 

Some philosophers have held that it is impossible to 
carry on reasoning without the use of language. The 
true nominalist went so far as to say that there are no 
such things as general noiions, and that general names 
therefore constitute all that is general in science and 



2ZS REQUISITES OF A [LEsa 

reasoning. Though this is no doubt false (see p. 13), it 
must nevertheless be allowed that unless general i^ieas 
were fixed and represented by words, we could never 
attain to sustained thought such as we at present enjoy» 
The use of language in the second purpose is doubtless 
indispensable in a practical point of view, and reasoning 
may almost be considered identical with the correct use 
of words. When language is used solely to assist reason- 
ing there is no need that the meaning of each word 
should be fixed ; we might use names, as the letters ;r, j/, 2^ 
a, b, r, &c., are used in algebra to denote any quantity 
that happens to occur in a problem. All that is requisite 
is never to confuse the meaning attributed to a word in 
one argument with the different meaning attributed in 
another argument. Algebra may, in fact, be said to con- 
sist of a language of a very perfect kind adapted to the 
second purpose only, and capable of leading a person to 
the solution of a problem in a symbolical or mechanical 
manner. 

Language, as it is furnished to us ready made by the 
habitual growth of centuries, is capable of fulfilling all 
three purposes, though by no means in a perfect manner. 
As words possess a more or less fixed customary meaning 
we can not only reason by their aid, but communicate our 
thoughts or record them ; and it is in this last respect we 
have now to treat the subject. 

The multitude of facts required for the establishment 
of a science could not be retained in the memory with 
sufficient accuracy. Hence an indispensable subsidiary 
of induction is the means of describing and recording our 
observations. Thus only can knowledge be accumulated, 
so that each observer shall start with the advantage of 
knowing what has been previously recorded and proved. 
It will be necessary then to consider the mode in which 
language serves for the registration of facts, and to investi- 



XXXIII.] PHILOSOPHICAL LANGUAGE. ' 289 

gate the requisite qualities of a philosophical language 
suitable to the needs of science. 

As an instrument of record language must evidently 
possess two principal requisites : 

1. Precision or definiteness of meaning. 

2. Completeness. 

A name is worse than useless unless, when used to 
record a fact, it enables us to ascertain what was the 
nature of the fact recorded. Accuracy and precision is 
then a more important quality of language than abun- 
dance. The want of an appropriate word will seldom 
give rise to actual error and fallacy ; it will merely oblige 
us to employ a circumlocutory phrase or else leave the 
fact unrecorded. But it is a self-evident convenience that 
whenever a thing, notion, or quality has often to be refer 
red to there should be a name appropriated to the 
purpose, and there ought only to be one name. Let us 
consider in succession what must be the character of a 
precise and complete language. 

It may not previously have struck the reader, but it is 
certainly true, that description is impossible without the 
assertion of resemblance between the fact described and 
some other fact. We can only describe a thing by giving 
it a name ; but how can we learn the meaning of that 
name ? If we describe the name by other names we only 
have more names of which the meanings are required. 
We must ultimately learn the meanings, not fro-m names 
but from things which bear those names. If anyone 
were ignorant of the meaning of blue he could not be in- 
formed but by reference to something that excited in him 
the sensation of blueness^ and had he been blind from 
birth he could not acquire any notion of what blueness 
was. There are indeed a number of words so familiar 
to us from childhood that we cannot tell when or how we 
learnt their meanings, though it must have been by refer- 

19 



290 REQUISITES OF A [less. 

ence to things. But when we come to the more precisQ 
use of names we soon have to make fresh reference to 
physical objects. Then we should describe the several 
kinds of blue colour as sky-blue, azure-blue, indigo-blue, 
cobalt-blue ; green colour we likewise distinguish as sea- 
green, olive-green, emerald-green, grass-green, &c. The 
shapes of leaves are described in Botany by such names 
as ovate, lanceolate, linear, pinnate, peltate, referring the 
mind respectively to an ^'gg^ a lance, a line, a feather, 
and a shield. In recording dimensions it is equally im- 
possible to avoid comparison with the dimensions of 
other things. A yard or a foot has no meaning unless 
there be a definite standard yard or foot which fixes its 
meaning ; and the reader is probably aware that when the 
physical standard of a length is once completely lost it 
can never be recovered. The word is nothing unless we 
somewhere have the thing to which it corresponds. 

The first requisite of a philosopMcal language evident- 
ly is that "every general name must have a certain and 
knowable meaning.'^ It need hardly be mentioned that 
singular or proper names, the names of distinct objects, 
must likewise be known; but as such names are merely 
marks imposed upon the things they do not need the 
same consideration. General names are a more difficult 
subject, because, as we have seen in Lesson v., they have a 
double meaning in denotation or extension, and connota- 
tion or intension. Of these two meanings the connotation 
is the one which must be fixed ; the other cannot as 
a general rule be limited and defined. Had the name 
planet been restricted to Jupiter, Saturn, Mars, Venus, 
and Mercury, the planets known before the invention of 
the telescope, we should have had to find a new name for 
those subsequently discovered, and should even then 
commit the fault of calling by different names those things 
which are closely similar. But if by planet we mean any 



XXXilL] PHILOSOPHICAL LANGUAGE, 291 

round body revolving round the sun in an orbit of slight 
elHpticity, it v/ill include all such bodies as may be dis 
covered from time to time, of which more than 100 are 
already known. Similarly locomotive engine is not merely 
the name of a number of engines now actually existing ; 
for if so a new name must be needed every week 
as some new engine is made or an old one destroyed. 
What is- fixed in a general name is its connotation, or the 
qualities irnplied in the things bearing the name. We 
ought therefore as far as possible to define the meaning 
of every general name we use, not by naming the objects 
which it denotes, but the qualities, which it connotes. 
Having however considered the subject of definition in 
previous Lessons (xii. and Xlll.), we need only inquire 
here how far it is desirable to employ words which are 
in current use in preference to newly invented terms. 

The advantage of an old term is that it possesses force 
of meaning for all persons, and so far saves the necessity 
of learning the meaning of a strange technical expression. 
Every one knows what heat is, and the expression science 
of heat bears meaning to every person however unlearned. 
But there is this objection against old terms to be noted, 
that they are almost always subject to ambiguity; accord- 
ingly it will be found that the scientific man really uses 
the word heat differently from other persons. All things 
are more or less hot in science, whereas in common life 
we could never say that ice was hot or contained heat. 
In fact heat means ordinarily the excess of temperature 
above the ordinary mean, and the notion is purely relative 
to that of cold. We also apply the word analogously to 
sensations of taste, as when we say pepper is hot, or 
even to purely mental phenomena, as in a hot dispute, a 
hot temper, &c. If to avoid these ambiguities we invent 
a new term, Caloric^ we may give it any precision of 
meaning we like, but we raise one more obstacle to the 

19—2 



292 REQUISITES OF A [LESS. 

study of science, because there is one more technical 
term to be learnt. 

This difficulty is especially great in the science of 
political economy. We there deal with such familiar 
ideas as wealth, money, value, currency, capital, labour, 
exchange, but it is the very familiarity of the ideas which 
occasions the greatest difficulty, because different people 
attach different meanings to the words, and infinite logo- 
7nachy (Greek \oyos, word; /xax^? battle), or disputes 
arising on merely verbal questions, is the result. Even if 
a writer carefully defines the meaning in which he uses 
those terms he cannot oblige other persons to bear the 
definitions in mind. The other alternative of inventing 
wholly new terms is out of the question, as it would un- 
doubtedly render a work intolerable to most readers. 
The only advice that can be given is to introduce a new 
term where it is likely to be readily accepted and to dis- 
place an old ambiguous term ; but otherwise to endeavour 
to remove the ambiguity of the old term by constantly 
keeping in view a precise definition of the intended 
meaning. 

A complete philosophical language will be composed 
of two distinct kinds of terms, which form respectively 
the descriptive terminology and the nomenclature of the 
science. 

A descriptive terminology, as pointed out by Dr 
Whewell, must include all the terms required to describe 
exactly what has been observed concerning any object or 
phenomenon, in order that we may possess a permanent 
record of the observation. For every quality, shape, 
circumstance, degree or quantity there must be an appro- 
priate nanle or mode of expression. Thus in recording 
the discovery of a new mineral we ought to be able to fix 
in words its exact crystalline form, its colour, its degree 
of hardness, its specific gravity, smell and taste ii any, 



XXXIII.] PHILOSOPHICAL LANGUAGE. 293 

and many other qualities which may possess importance. 
Modern botany arose from the efforts of Linnasus to 
create a system of terms by which every part and 
character of a plant can be accurately described. The 
language of botany, as since improved, presents the most 
complete instance of a scientific terminology. Geology 
suffers much, as I apprehend, from the difficulty of find- 
ing accurate terms ; such names as trap, basalt, gneiss, 
granite, tuff, greenstone, trachyte, porphyry, lava, &c., 
are exceedingly vague and almost impossible to define, 
and at the same time to distinguish. Where a quality 
does not admit of degree or quantity it only requires a 
single name ; otherwise we must find some mode of exact 
measurement and expression. The invention of any in- 
strument for measuring a quality which has been before 
unmeasured is always an important step in science, and 
the construction of the thermometer by Fahrenheit and the 
pendulum clock by Huyghens were great eras in science. 

On the other hand, each science requires a nomen- 
clature or collection of names for the distinct objects or 
classes of objects treated in it. In mineralogy the names 
of separate minerals, such as haematite, topaz, amphibole, 
epidote, blende, polybasite, form the nomenclature ; in 
chemistry we have all the names of the elements, together 
with a vast apparatus of names for organic and other 
compounds, such as ethyl, acet)l, cyanogen, napthalin, 
benzol, &:c. In astronomy the names of the planets, 
satellites, nebulse, constellations or individual stars, form 
a nomenclature of by no means a perfect or convenient 
kind ; and geology has similarly a nomenclature neces- 
sarily of an incomplete character, in the names of the 
successive formations, silurian, devonian, carboniferous, 
permian, triassic, eocene, miocene, pliocene, post-plio- 
cene, &c. 

It is evident that a nomenclature must possess names 



294 REQUISITES OF A [less, 

of various degrees of generality, including individual 
objects if they need separate record, infijncB species if 
3uch there be, with wider classes, up to the stimnia 
genera^ or widest notions embraced in the science. In 
astronomy we deal chiefly with the names of individual 
objects, and there is as yet but little scope for classi- 
fication. In such natural sciences as botany or zoology 
there is seldom or never any need of names for indi- 
viduals, as an indefinite multitude of individuals generally 
resemble each other very closely in a great number of 
properties, so as to constitute what has been called a 
natural kind. Mr Mill uses this term to denote " one of 
those classes which are distinguished from all others, not 
by one or a few definite properties, but by an unknown 
multitude of them ; the combination of properties on 
which the class is grounded being a mere index to an 
indefinite number of other distinctive attributes." 

According to Mr Mill's language he seems to include 
in a nomenclature only the names of supposed species; 
for he says : — "A nomenclature maybe defined, the collec- 
tion of names of all kinds with which any branch of 
knowledge is conversant ; or more properly, of all the 
lowest kinds, or ififimce species^ those which may be sub- 
divided indeed, but not into kinds, and which generally 
accord with what in natural history are termed simply 
species." But the fact is that naturalists have now aban- 
doned the notion that the species is any definite form ; 
many species are divided already into subspecies and 
varieties, or even varieties of varieties; and according to 
the principles of Darwin's theory the subdivision might 
go on indefinitely. It is surely most reasonable to regard 
the natural kingdoms of vegetables and animals as ar- 
ranged in an indefinite series of classes and subclasses, 
and all the names attaching to any such classes belong 
to the nomenclature. 



XXXIII.] PHILOSOPHICAL LANGUAGE, 295 

Again, Mr Mill does not inckide in the nomenclature 
such general names as denote conceptions artificially 
formed in the course of induction and investigation. Ac- 
cordingly, besides a terminology suited for describing 
with precision the individual facts observed, there is a 
branch of language containing " a name for every com- 
mon property of any importance or interest, which we 
detect by comparing those facts : including (as the con- 
cretes corresponding to those abstract terms) names for 
the classes which we artificially construct in virtue of 
those properties, or as many of them, at least, as we have 
frequent occasion to predicate any thing of.'* As exam- 
ples of this class of names he mentions Circle, Limit, 
Momentum, Civilization, Delegation, Representation. 
While the nomenclature contains the names of natural 
classes, this third branch of language would apparently 
contain the names of artificial ideas or classes. 

But I feel great difficulty in giving a clear account of 
Mr Mill's views on this subject, and, as my object in these 
Lessons does not allow of the discussion of unsettled 
questions, I must conclude by referring the reader who 
desires to continue the subject, to the 4th and 6th chap- 
ters of the 4th Book of Mr Mill's System of Logic ^ which 
treat of the Requisites of a P'hilosophical Language, 

See Dr Whewell's " Aphorisms concerning the Lan- 
guage of Science," at the end of his Philosophy oj 
the Inductive Sciences, 

Thomson's Outline of the Laws of Thought^ con- 
tains most interesting remarks on the general nature 
and use of Language, §§ 17 — ^31. 



QUESTIONS AND EXERCISES. 

Lesson I. — Introduction, 

1. What are the meanings of a Law of Nature, and a 

Law of Thought ? 

2. Explain the distinction between the Form of 

Thought, and the Matter of Thought. 

3. In what sense may Logic be called the Science of 

Sciences ? 

4. What is the derivation of the name Logic ? 

5. How does a Science differ from an Art, and why is 

Logic more in the form of a Science than an 
Art ? 

6. Can we say that Logic is a necessary aid in correct 

reasoning, when persons who have never studied 
logic reason correctly ? 

Lesson IL — Three Parts of Logic, 

1. Name the parts of which a syllogism is composed. 

2. How far is it correct to say that Logic is concerned 

with language ? 

3. What are the three acts of mind considered in 

Logic ? Which of them is more especially the 
subject of the Science ? 

4. Can you state exactly what is meant by a general 

notion, idea, or conception ? 

5. How do the Nominalists, Realists, and Concep- 

tualists differ in their opinions as to the nature 
of a general notion ? 
6 What is the supposed fourth part of Logic? 



QUESTIONS AND EXERCISES. 2t^3 

Lesson IIL — Terms, 

1. Define a name or term. 

2. What is a categorematic term ? 

3. Explain the distinction between a collective and a 

general term. 

4. Distinguish the collective and distributive use of 
the word all in the following : — 

(i) Non omnis moriar {j.e, I shall not all die). 

(2) " All men find their own in all men's good, 

And all men join in noble brotherhood." 

Tennyson, 

(3) Non omnia possumus omnes (/. e, we cannot all 

do all things). 

5. Which of the following are abstract terms ? 

Act, ingratitude, home, hourly, homeliness, intro- 
duction, individuality, truth, true, trueness, 
yellow, yellowness, childhood, book, blue, in- 
tention, reason, rationality, reasonableness. 

6. Define a negative term, and mention the mark by 

which you may recognise it. 

7. Distinguish a privative from a negative term, and 

find some instances of privative terms. 

8. Describe the logical characters of the following 

terms, with the precautions given at p. 26. 



Metropolijs 
Book 


Consciousness 
Lord Chancellor 


Sect 

Nation 


Library 
Great Britain 


Vegetable Kingdom 
Brilliance 


Institution 
Light 


Caesar 


Weight 


Observation 


Void 
Gold 


Sensation 
Caesar 


Tongue 
Air 


Prime Minister 


Csesarism 


Mentor 


Indigestibility 
Manchester 


Application 
Individual 


Anarchy 
Retribution 


Recollection 


Volume 


Solemnity 



298 QUESTIONS AND EXERCISES. 



Insignificant 

Brilliant 

Independence 

Heaviness 

Illustration 

Section 

Whiteness 



Lar^guage 

Adornment 

Agreement 

Obliquity 

Motionless 

Henry VIH. 

Formal Logic 



Understanding 

Geology 

Demeanour 

Resemblance 

Departure 

Nestor 

Alexander 



Lesson IV. — Ambiguity of Terms, 

1. Define uni vocal terms, and suggest some terms 

whicii. are perfectly univocal. 

2. What are the other names by which equivocal 

terms are often called ? 

3. Distinguish the three kinds of ambiguous terms, 

and find instances of each. 

4. Distinguish the three causes by which the third and 

most important class of ambiguous terms have 
been produced. 

5. Explain the ambiguity of any of the following 

terms, referring each to its proper cause, and 
tracing out as far as possible the derivation of 
each separate meaning from the original meaning. 



Bill 


Minister 


Subject 


Letter 


Table 


Clerk 


Object 


Star 


Term 


Order 


Earth 


Pole 


School 


Wood 


Law 


Reason 


Air 


Bull 


Sensation 


Bed 


Glass 


Volume 


Art 


Bowl 


Peer 


Scale 


Interest 


End 


Sense 


Feeling 


Paper 


Division 


Ball 


Kind 


Bolt 


Class 



Lesson V .—Twofold meaning of Terms, 

I. Distinguish very carefully the meanings in ex- 
tension and intension of the terms — 
Quadruped, railway, human being, engine, moun- 
tain, Member of Parliament. 



QUESTIONS AND EXERCISES, 299 

2. Enumerate the synonyms or other names used 

instead of extension and intension. 

3. According to what law is the quantity of extension 

connected with the quantity of intension ? Show 
that the law holds true of the following series of 
terms — 
(i) Iron, metal, element, matter, substance. 

(2) Matter, organized matter, animal, man. 

(3) Ship, steamship, screw-steamship, iron screw- 

steam-ship, British iron screw steamship. 

(4) Book, printed book, dictionary, Latin dic- 

tionary. 

4. Distinguish between the connotation and deno- 

tation of a term. 

5. Select from the list of terms under Lesson ill., 

Question 8 (p. 297), such terms as are non-con- 
notative according to Mr Mill's views. 

6. Arrange the following terms in series as in ques- 

tion 3, placing each term of greater extension 
before a term of less extension. Point out 
which are the terms of greatest and least inten- 
sion in each series. 



Emperor 


Animal 


Planet 


Teacher 


Dissenter 


Mammalian 


Baptist 


Individual 


Matter 


Timber 


Jupiter 


Solicitor 


Person 


Ruler 


Quadruped 


Horse 


Organized substance 


Being 


Heavenly body 


Lawyer 


Napoleon III. 


Christian 


Alexander 


Episcopalian 



Lesson VI. — Growth of Language, 

I. Trace out the generalization or specialization which 
has taken place in any of the following words : — 



3O0 QUESTIONS AND EXERCISES. 

Kind, genus, class, species, order, rank, Augustus, 
president, speaker, Utopia, rock, CommoiiSj"' 
doctor. 

2. Point out metaphors derived from the notions of 

weight, straightness, rock, wind. 

3. Distinguish as accurately as possible the meanings 

of the following synonyms : — 
Sickness, malady ; mud, mire ; confutation, refu- 
tation ; boundary, limit ; mind, intellect ; recol- 
lection, reminiscence; procrastination, dilato- 
riness ; converse, reverse, obverse, inverse. 
4* Form lists of all the words derived from any of the 
following roots : — 
(i) Te7idere^ to stretch, as in intention, attention, 

(2) Ponere^ to place, as in position, supposition. 

(3) Genus^ tribe or kind, as in genus, generation. 

(4) Munus, gift, as in remuneration, common (Latin, 

Communis), 

(5) Modus^ shape or fashion, as in mood, moderate, 

(6) ScriberCy to write, as in scribe, inscription, de- 

scribe. 

(7) Capere to take, as in deception, incipient 

Lesson VI L — Leibnitz on Knozu ledge, 

1. What are tha characters of perfect knowledge? 

2, Describe the character of the knowledge which we 

have of the following notions or objects : — 
A syllogism. 
Electricity. 
Motion. 
A triangle. 
Eternity. 

The weight of the earth (5852 trillions of tons) 
The colour of the sky. 



QUESTIONS AND EXERCISES, 301 

3. Explain exactly what you mean by intuitive know- 
ledge. 

Lesson VIII. — Propositions, 

1. Define a proposition, and name the parts of which 

it is composed. 

2. How are propositions classified? 

3. Name the four kinds of categorical propositions, 

and their symbols, 

4. Under which classes are singular and indefinite 

propositions placed ? 

5. Enumerate the most usual signs of the quantity of 

a proposition. 

6. What are modal propositions according to early 

logicians, and according to Thomson 1 

7. How far do logicians consider propositions with 

regard to their truth or falsity ? 

Lesson IX. — Opposition of Propositiotis, 

1. State the quantity of the subject and predicate in 

each of the propositions A, E, I, 0. 

2. Select out of the following propositions, pairs of 

contrary, contradictory, subaltern, and subcon- 
trary propositions : — 

(1) Some elements are known, 

(2) No elements are known. 

(3) All elements are known. 

(4) Not all elements are known. 

(5) Some elements are not known. 

(6) All elements are not known. 

3. What propositions are true, false, or doubtful, 

(i) when A is false, (3) when I is false, 

(2) when E is false, (4) when is false.? 

4- Prove by means of the contradictory propositions 



502 QUESTIONS AND EXERCISES. 

that subcontrary propositions cannot both be 
false. 

5. Show by means of the subcontrary propositions 

that contrary propositions may both be false. 

6. What quantity would you assign to each of the 

following propositions ? 
(i) Knowledge is power. 

(2) Nebulse are material bodies. 

(3) Light is the vibration of an ether. 

(4) Men are more to be trusted than we think. 

(5) The Chinese are industrious. 

7. Why is it desirable in controversy to refute a state- 

ment by its contradictory and not by its contrary? 



Lksson X. — Conversion and Im7nediate Inference. 

1. Define inference and conversion. 

2. What are converse and convertend propositions ? 

3. State the rules of valid conversion. 

4. Name all the kinds of conversion. 

5. By what process do we pass from each of the fol- 

lowing propositions to the next ? 
(i) No knowledge is useless. 

(2) No useless thing is knowledge. 

(3) All knowledge is not useless. 

(4) All knowledge is useful. 

(5) What is not useful is not knowledge. 

(6) What is useless is not knowledge. 

(7) No knowledge is useless. 

6. Give the logical opposites of the following propo% 

sition, and the converse of its contradictory :— » 
*^ He cannot become rich who will hot labour.'' 
7 Apply negative conception to the proposition ^^ All 
men are fallible ;" then convert and show that 
the result is the contrapositive of the original 



QUESTIONS AND EXERCISES. 303 

8. Classify the propositions subjoined into the foul 
following groups: — 
a. Those which can be inferred from (i). 
b» Those from which (i) can be inferred. 

c. Those which do not contradict (i), but cannot 

be inferred from it. 

d. Those which contradict (i). 

(i) All just acts are expedient acts. 

(2) No expedient acts are unjust. 

(3) No just acts are inexpedient. 

(4) All inexpedient acts are unjust. 

(5) Some unjust acts are inexpedient. 

(6) No expedient acts are just. 

^ (7) Some inexpedient acts are unjust. 

(8) All expedient acts are just. 

(9) No inexpedient acts are just. 

(10) All unjust acts are inexpedient. 

(11) Some inexpedient acts are just acts. 

(12) Some expedient acts are just. 

(13) Some just acts are expedient. 

(14) Some unjust acts are expedient. 

Lessons VIII. IX. and X. — Examples of Propositions^ 

The reader is desired to ascertain the logical character 
of each of the following propositions; he is to state of 
each whether it is affirmative or negative, universal, p^.r- 
ticular, singular or indefinite, pure or modal, exclusive or 
exceptive, &c. ; when irregularly stated he is to reduce the 
proposition to the simple logical order; he is then to 
convert the proposition, and to draw immediate inferences 
from it by any process which may be applicable. 

(i) All birds are feathered. 

(2) No reptiles are feathered. 

(3) Fixed stars are self-luminous. 



304 QUESTIONS AND EXERCISES. 

(4) Perfect happiness is impossible. 

(5) Life every man holds dear. 

(6) Every mistake is not a proof of ignorance. 

(7) Some of the most valuable books are seldom read* 

(8) He jests at scars who never felt a wound. 

(9) Heated metals are softened. 

(10) Not one of the Greeks at Thermopyl^ escaped. 
(u) Few are acquainted with themselves. 

(12) Whoso loveth instruction loveth knowledge. 

(13) Nothing is harmless that is mistaken for a virtue. 

(14) Some of our muscles act without volition. 

(15) Metals are all good conductors of heat. 

(16) Fame is no plant that grows on mortal soil. 

(17) Only the brave deserve the fair. 

(18) No one is free who doth not command himself. 

(19) Nothing is beautiful except truth. 

(20) The wicked shall fall by his own wickedness. 

(21) Unsafe are all things unbecoming. 

(22) There is no excellent beauty that hath not some 

strangeness in the proportion. 

(23) It is a poor centre of a man's actions, himself. 

(24) Mercy but murders, pardoning those that kilL 

(25) I shall not all die. {Non omnis moriar,) 

(26) A regiment consists of two battalions. 

(27) 'Tis cruelty to load a falling man. 

(28) Every mistake is not culpable. 

(29) Quadrupeds are vertebrate animals. 

(30) Not many of the metals are brittle. 

(31) Many are the deserving men who are unfortunate. 

(32) Amalgams are alloys of mercury. 

(33) One kind of metal at least is liquid. 

(34) Talents are often misused. 

(35) Some parallelograms have their adjoining sides 

equal. 

(36) Britain is an island. 

(37) Romulus and Remus were twins. 



QUESTIONS AND EXERCISES. 305 

(38) A man's a man. 

(39) Heaven is all mercy. 

(40) Every one is a good judge of his own interests. 

(41) A-11 parallelograms have their opposite angles equal 

(42) Familiarity breeds contempt. 

(43) No one is always happy. 

(44) Every little makes a mickle. 



Lesson XI. — Logical Analysts of SenUnces 

1. How does the grammatical predicate differ from the 

logical predicate ? 

2. Distinguish between a compound and a complex 

sentence ; and between coordinate and subordinate 
propositions. 

3. Enumerate the grammatical expressions which may 

form 
(i) A subject. (4) An object. 

(2) An attribute. (5) An adverbial. 

(3) A predicate. 

4. Examine the following sentences, ascertain which 

are compound or complex, and point out the co- 
ordinate or subordinate propositions, 
(i) Happy is the man that findeth wisdom, and the 
man that getteth understanding. 

(2) Heat, being motion, can be converted into me- 

chanical force. 

(3) Ceres, Pallas, Juno, and Vesta are minor planets, 

or asteroids. 

(4) Knowledge comes, but wisdom lingers. 

(5) Fortune often sells to the hasty what she gives to 

those who wait. 

(6) Thousands at His bidding speed, 
And post o'er land and ocean without rest ; 
They also serve who only stand and wait. 

20 



3o6 QUESTIONS AND EXERCISES. 

if) Pride that dines on vanity, sups on contempt 

(8) Nobody can be healthful without exercise, neither 

natural body, nor politic. 

(9) Nature is often hidden, sometimes overcome, 

seldom extinguished. 

(10) It is impossible to love and be wise. 

(11) Though gods they were, as men they died. 

(12) He that is not industrious envieth him that is. 

(13) Ye are my friends, if ye do whatsoever I command 

you. — ^John xv. 14. 

(14) The wisdom that is from above is first purej then 

peaceable, gentle, and easy to be intreated, 
full of mercy, and good fruits, without par- 
tiality, and without hypocrisy. — ^James iii. 17. 
5. Analyse in the form of a scheme or diagram any of 
the following sentences : — 
(i) The first aphorism of Bacon's A^<9^/^^ Organum, 
on p. 229. 

(2) Some judgments are merely explanatory of their 

subject, having for their predicate, a conception 
which it fairly implies, to all who know and can 
define its nature. 

(3) There be none of the affections which have been 

noted to fascinate or bewitch, but love and 

envy : they both have vehement wishes ; they 

frame themselves readily into imaginations and 

suggestions ; and they come easily into the eye, 

especially upon the presence of the objects, 

which are the points that conduce to fascination, 

if any such there be. 

Further examples for analysis must be sought in 

Dalgleish's Grammatical Analysis^ with Progressiva Ex- 

trcises. (Oliver and Boyd.) Edinburgh, 1866. Price 9^. 



QUESTIONS AND EXERCISES, 



y>7 



Lesson XII. — T/^e Predicables^ etc, 

!• Define each of the five predicables. 

2. In what sense may we say that the genus is part of 

the species, and in what sense that the species is 
part of the genus ? 

3. Select from the terms in the 6th Question of Les- 

son v., p. 299, such as are genera, species, 
highest genera, or lowest species of other terms. 
4- Explain the expressions sui generis, homogeneous, 
heterogeneous, s'ummum genus, infima species, 
tree of Porphyry. 

5. N ame a property and accident of each of the follow- 

ing classes : — Circle, Planet, Bird, Member of 
Parliament, Ruminant Animal. 

6. What are the rules of correct logical division. 

7. The first name in each of the following series of 

terms is that of a class which you are to divide 
and subdivide so as to include all the subjoined 
minor classes in accordance with the laws of 
division. 



(i1 People, 


(2) Trlajzgle, 


(3) Reasoning, 


Laity 


Equiangular 


Induction (Imperfect) 


Aliens 


Isosceles 


Deduction 


Nati^ralized 


Right-angled 


Mediate Inference 


Subjects 


Scalene 


Induction 


Peers 


Obtuse-angled 


Hypothetical Syllogism 


NatTiral-bom 




Disjunctive Syllogiom 


Subjects 






Clergy 






Baronets 






Commons 







8. Divide any of the following classes : — Governments, 

Sciences, Logical terms, Propositions. 

9. Of what does a logical definition consist ? 

20 — i 



3o8 QUESTIONS AND EXERCISES. 

10. What are the rules of correct definition ? 

1 1. What rules do the following definitions break ? 
(i) Life is the sum of the vital functions. 

(2) Genus is the material part of the species. 

(3) Illative conversion is that in which the truth of 

the converse can be inferred from that of the 
convertend. 

(4) Mineral substances are those which have not 

been produced by the powers of vegetable or 
animal life. 

(5) An equilateral triangle 'is a triangle whose sides 

and angles are respectively equal. 

(6) An acute-angled triangle is one which has an 

acute angle. 

Lesson XIII. — Pascal and Descartes on Method, 

(i) What is the use of nominal definitions ? 

(2) How must we employ definitions in order to avoid 

confusion ? 

(3) How far can we be said to be free to use any name 

for any object 1 

(4) What according to Pascal is the true method of 

avoiding error ? 

(5) How do we learn the meanings of words which 

cannot be defined 1 

(6) Give instances of words which can be clearly de- 

fined and of others which cannot. 

(7) State the five rules of method given in the Port 

Royal Logic. 

(8) Explain Descartes' rules for the attainment of 

truth. 

Lesson XIV. — Laws of Thought. 

&. State the three Fundamental Laws of Thcu^bt^ and 
apply them to the following notions : — 



QUESTIONS AND EXERCISES. 309 

(i) Matter, organic, inorganic. 

(2) Undulations, polarized, non -polarized. 

(3) Figure, rectilinear, curvilinear. 

2. Is it wrong to assert that animal cannot both be 
vertebrate and invertebrate, seeing that some 
animals are vertebrate and some are not ? 

5, Select from the following such terms as are nega- 
tives of the others, and such as are opposites : — ■ 
Light, plenum, gain, heat, decrease, loss, darkness, 
cold, increase, vacuum. 

4. How is Aristotle's dictum applicable to the follow- 
ing arguments? 

(i) Silver is a good conductor of electricity ; for such 

are all the metals. " 
(2) Comets cannot be without weight ; for they are 

composed of matter, which is not without weight. 

Lesson XV. — Syllogism: the Rules, 

1. Distinguish mediate and immediate inference. 

2. Define syllogism, and state with what it is synony- 

mous. 

3. What are the six principal and two subordinate 

rules of the syllogism.'^ 

4. In the following syllogisms point out in succession 

the conclusion, the middle term, the major term; 
the minor term, the major premise and the minor 
premise, observing this precise order. 

(i) All men are fallible ; 
All kings are men ; 
Therefore all kings are fallible. 
(2) Platinum is a metal ; 

All metals combine with oxygen ; 
Therefore Platinum combines with oxygen. 



3IO QUESTIONS AND EXERCISES. 

(3) Hottentots are capable of education ; for Hotten* 
tots are men, and all men are capable of edu- 
cation. 
5. Explain carefully what is meant by non-distribution 
of the middle term. 

Lesson XVI.~rA^ Moods and Figures of the 

Syllogism. 

1. Name the rules of the syllogism which are broken 

by any of the following moods, no regard being 
paid to figure : — 
AIA, EEI, lEA, lOI, IIA, AEI. 

2. Write out all the 64 moods of the syllogism and 

strike out the 53 invalid ones. 

3. Show in what figures the following premises give a 

valid conclusion : — A A, A I, EA, OA. 

4. In what figures are I E O and E I O valid ? 

5. To what moods do the following valid syllogisms 

belong ? Arrange them in correct logic;J order. . 
(i) Some Y's are Z's. (2) All Z's are Y's. 

No X's are Y's. No Y's are X's. 

Some Z's are not X's. No Z's are X'3. 

(3) No fish suckles its young ; 
The whale suckles its young ; 
Therefore the whale is no fish. 

6. Deduce conclusions from the following premises ] 

and state to what mood the syllogism beloncrs, 

(1) Some amphibious animals are mammalian. 
All mammalian animals are vertebrate. 

(2) All planets are heavenly bodies. 
No planets are self-luminous. 

(3) Mammalian animals are quadrupeds. 
No birds are quadrupeds. 

(4) Ruminant animals are not predacious. 
The lion is predacious. 



QUESTIONS AND EXERCISES. 311 

7. Invent examples to show that false premises may 

give true conclusions. 

8. Supply premises to the following conclusions : — 
(i) Some logicians are not good reasoners. 

(2) The rings of Saturn are material bodies. 

(3) Party government exists in every democracy. 

(4) All fixed stars obey the law of gravitation. 

Lesson XVII.— Z%<? Syllogism; Reduction, 

1. State and explain the mnemonic lines Barbara, 

Celarent, &c. 

2. Construct syllogisms in each of the following moods, 

taking X, Y, Z, for the major, middle, and minor 
terms respectively, and show how to reduce them 
to the first hgure : — 
Cesare, Festino, Darapti, Datisi, Ferison, Camenes, 
Fesapo. 

3. What is the use of Reduction ? 

4. Prove that the following premises cannot give a 

universal conclusion — E I, I A, O A, I E. 

5. Prove that the third figure must have an affirmative 

minor premise, and a particular conclusion. 

6. Reduce the moods Cesare and Camenes by the 

Indirect method, or Reductio ad Impossibile. 

Lesson XVIII. — Irregular and Compound Syllogisms. 

1. Describe the meaning of each of the terms — En- 

thymeme, Prosyllogism, Episyllogism, Epichei- 
rema, Sorites. 

2. Make an example of a syllogism in which there are 

two prosyllogisms. 

3. Construct a sorites of four premises and resolve it 

into distinct syllogisms. 

4. What are the rules to which a sorites must conform? 



312 QUESTIONS AND EXERCISES. 

5. The reader is requested to analyse the following 
arguments, to detect those which are false, and to 
ascertain the rules of the syllogism which they 
break ; if the argument appears valid he is to 
ascertain the figure and mood to which it belongs, 
to state it in correct logical form, and then if it be 
in an imperfect figure to prove it by reduction to 
the first figure. The first six of the examples 
should be arranged both in the extensive and 
intensive orders. 

1. None but mortals are men. 

Monarchs are men. 

Therefore monarchs are mortals. 

2. Personal deformity is an affliction of nature. 

Disgrace is not an affliction of nature. 
Therefore personal deformity is not disgrace. 

3. Some statesmen are also authors; for such are 

Mr Gladstone, Lord Derby, Lord Russell, and 
Sir G. C. Lewis. 

4. This explosion must have been occasioned by gun- 

powder; for nothing else would have possessed 
sufficient force. 

5. Every man should be moderate; for excess will 

cause disease. 

6. Blessed are the merciful; for they shall obtain 

mercy. 

7. As almost all the organs of the body have a 

known use, the spleen must have some use. 

8. Cogito, ergo sum. (I think, therefore I exist.) 

9. Some speculative men are unworthy of trust ; for 

they are unwise, and no unwise man can be 
trusted. 
10. No idle person can be a successful writer of liis- 
tory ; therefore Hume, Macaulay, Hallam and 
Grote must have been industrious. 



QUESTIONS AND EXERCISES. 313 

11. Who spareth the rod, hateth his child; the parent 

who loveth his child therefore spareth not the 
rod. 

12. Comets must consist of heavy matter; for other- 

wise they would not obey the law of gravitation. 

13. Lithium is an element; for it is an alkali-pro- 

ducing substance, which is a metal, which is 
an element. 

14. Rational beings are accountable for their actions ; 

brutes not being rational, are therefore exempt 
from responsibility. 

15. A singular proposition is a universal one; for 

it applies to the whole of its subject 

16. Whatever tends to withdraw the mind from pur- 

suits of a low nature deserves to be promoted ; 
classical learning does this, since it gives us 
a taste for intellectual enjoyments; therefore it 
deserves to be promoted. 

17. Bacon was a great lawyer and statesman ; and as 

he was also a philosopher, we may infer that any 
philosopher may be a great lawyer and statesman. 

18. Immoral companions should be avoided ; but some 

immoral companions are intelligent persons, so 
that some intelligent persons should be avoided. 

19. Mathematical study undoubtedly improves the 

reasoning powers; but, as the study of logic is 
not mathematical study, we may infer that it does 
not improve the reasoning powers. 

20. Every candid man acknowledges merit in a rival ; 

every learned man does not do so; therefore 
every learned man is not candid. 

Lesson XIX. — Conditional Arguments, 

I. What are the kinds of conditional propositions, 
and by what signs can you recognise them.? 



314 QUESTIONS AND EXERCISES. 

2. What are the rules of the hypothetical syllogism ? 

3. To what categorical fallacies do breaches of these 

rules correspond? 

4. Select from the following such as are valid argu- 

ments, and reduce them to the categorical form ; 
explain the fallacious reasoning in the others, 
(i) Rain has fallen if the ground is wet ; but the 
ground is not wet ; therefore rain has not fallen. 

(2) If rain has fallen, the ground is wet ; but rain has 

not fallen ; therefore the ground is not wet. 

(3) The ground is wet, if rain has fallen ; the ground 

is wet ; therefore rain has fallen. 

(4) If the ground is wet, rain has fallen ; but rain has 

fallen ; therefore the ground is wet 
N. B. In these as in other logical examples the 
student must argue only from the premises, and not from 
any other knowledge of the subject-matter. 

5. Show that the canons of syllogism (p. 121) may 

be stated indifferently in the hypothetical or 
categorical form. 

6. State the following in the form of a Disjunctive or 

Dilemmatic argument, and name the kind to 
which it belongs. 
If pain is severe it will be brief; and if it last long it 
will be slight; therefore it is to be patiently borne. 

Lessons XX. and XXI — Fallacies, 

1. Classify fallacies. 

2. Explain the following expressions : 

A dicto secundum quid ad dictum simpliciter ; igno- 
ratio eienchi; argumentum ad hominem; argu- 
mentum ad populum ; petitio principii ; circulus 
in probando; non sequitur; post hoc erga 
propter hoc 



QUESTIONS AND EXERCISES. 315 

3. What is arguing in a circle j and what is a ques- 

tion-begging epithet? 

4. What differences of meaning may be produced in 

the following sentence by varying the accent? 

" Newton's discovery of gravitation is not generally 
believed to have been at all anticipated by 
several philosophers in England and Holland." 
$. Point out the misinterpretations to which the fol- 
lowing sentences might be liable. 

(i) He went to London and then to Brighton by 
the express train. 

(2) Did you make a long speech at the meeting? 

(3) How much is five times seven and nine ? 



MISCELLANEOUS EXAMPLES. 
Lessons IX. to XXI. 

{Continued from p. 313.) 

The following examples consist partly of true and 
partly of false arguments. The reader is requested to 
treat them as follows : 

1. If the example is not in a simple and complete 

logical form, to complete it in the form which 
appears most appropriate. 

2. To ascertain whether it is a valid or fallacious 

argument. 

3. To assign the exact name of the argument or fal- 

lacy as the case may be. 

4. If a categorical syllogism, to reduce it to the first 

figure. 
' 5. If a hypothetical syllogism, to state it in the cate- 
gorical form. 
21, Elementary substances alone are metals. Iron is 
a metal ; therefore it is an elementary substance. 



3i6 QUESTIONS AND EXERCISES. 

22. No Athenians could have been Helots ; for all the 

Helots were slaves, and all Athenians were free 
men. 

23. Aristotle must have been a man of extraordinary 

industry; for only such a man could have pro- 
duced his works. 

24. Nothing is better than wisdom; dry bread is 

better than nothing ; therefore dry bread is better 
than wisdom. 

25. Pitt was not a great and useful minister; for 

though he would have been so had he carried 
out Adam Smith's doctrines of Free Trade, he 
did not carry out those doctrines. 

26. Only the virtuous are truly noble ; some who are 

called noble are not virtuous; therefore some 
who are called noble are not truly noble. 

27. Ireland is idle and therefore starves ; she starves, 

and therefore rebels. 

28. No designing person ought to be trusted; en- 

gravers are by profession designers; therefore 
they ought not to be trusted. 

29. Logic as it was cultivated by the schoolmen 
, proved a fruitless study ; therefore Logic as it is 

cultivated at the present day must be a fruitless 
study likewise. 

30. Is a stone a body? Yes. Then is not an animal 

a body? Yes. Are you an animal ? I think so. 
Ergo, you are a stone, being a body. — Lucian, 

31. If ye were Abraham's children, ye would do the 

works of Abraham. — John viii. 39. 

32. He that is of God heareth God's words : ye there- 

fore hear them not, because ye are not of God. 
— John viii. 47. 
83. Mahomet was a wise lawgiver ; for he studied the 
character of his people. 



QUESTIONS AND EXERCISES. 317 

S4. Every one desires virtue, because every one 
desires happiness. 

35. His imbecility of character" might have been in- 

ferred from his proneness to favourites ; for all 
weak princes have this failing. — De Morgan. 

36. He is brave who conquers his passions ; he who 

resists temptation conquers his passions; so that 
he who resists temptation is brave. 

37. Suicide is not always to be condemned ; for it is 

but voluntary death-, and this has been gladly 
embraced by many of the greatest heroes of 
antiquity. 

38. Since all metals are elements, the most rare of all 

the metals must be the most rare of all the 
elements. 

39. The express train alone does not stop at this sta- 

tion ; and as the last train did not stop it must 
have been the express train. 

40. Peel's remission of taxes was beneficial ; the taxes 

remitted by Peel were indirect; therefore the 
remission of indirect taxes is beneficial. ' 

41. Books are a source both of instruction and amuse- 

ment ; a table of logarithms is a book ; there- 
fore it is a source both of instruction and amuse- 
ment. 

42. All desires are not blameable ; all desires are liable 

to excess ; therefore some things liable to excess 
are not blameable. 

43. Whosoever intentionally kills another should suffer 

death ; a soldier, therefore, who kills his enemy 
should suffer death. 

44. Projectors are unfit to be trusted; this man has 

formed a project; therefore he rs unfit to be 
trusted. 
46. Few towns in the United Kingdom have more than 



3i8 QUESTIONS AND EXERCISES. 

300,000 inhabitahts ; and as all such towns ought 
to be represented by three members in Parlia- 
ment, it is evident that few towns ought to have 
three representatives. 

46. All the works of Shakspeare cannot be read in 

a day; therefore the play of Hamlet, being one 
of the works of Shakspeare, cannot be read in 
a day. 

47. In moral matters we cannot stand still ; therefore 

he who does not go forward is sure to fall behind. 

48. The people of the country are suffering from famine ; 

and as you are one of the people of the country 
• you must be suffering from famine. 

49. Those substances which are lighter than water 

can float upon it ; those metals which can float 
upon it are potassium, sodium, lithium, &c. ; 
therefore potassium, sodium, lithium, &c., are 
lighter than water. 

60. The laws of nature must be ascertained by De- 

duction, Traduction or Induction; but the former 
two are insufficient for the purpose ; therefore 
the laws of nature must be ascertained by In- 
duction. 

61. A successful author must be either very industrious 

or very talented ; Gibbon was very industrious, 
therefore he was not very talented. 

62. You are not what 1 am ; I am a man ; therefore 

you are not a man. 

63. The holder of some shares in a lottery is sure to 

gain a prize ; and as I am the holder of some 
shares in a lottery I am sure to gain a prize. 
54. Gold and silver are wealth ; and therefore the 
diminution of the gold and silver in the country 
by exportation is the diminution of the wealth 
' of the country. 



QUESTIONS AND EXERCISES, 3^9 

65. Over credulous persons ought never to be believed ; 

and as the Ancient Historians were in many 
instances over credulous they ought never to be 
believed. 

66. Some mineral compounds are not decomposed by 

heat ; all organic substances are decomposed by 
heat; therefore no organic substances are mi- 
neral compounds. 

57. Whatever schools exclude religion are irreligious ; 

Non-sectarian schools do not allow the teaching 
of religious creeds ; therefore they are irreligious. 

58. Night must be the cause of day ; for it invariably 

precedes it. 

59. The ancient Greeks produced the greatest master- 

pieces of eloquence and philosophy; the Lace- 
daemonians were ancient Greeks ; therefore they 
produced the greatest masterpieces of eloquence 
and philosophy. 

60. All presuming men are contemptible; this man, 

therefore, is contemptible ; for he presumes to 
believe his opinions are correct. 

61. If a substance is solid it possesses elasticity, and 

so also it does if it be liquid or gaseous ; but all 
substances are either solid, liquid or gaseous ; 
therefore all substances possess elasticity. 

62. If Parr's life pills are of any value those who take 

them will improve in health ; now my friend who 
has been taking them has improved in health ; 
therefore they are of value. 

63. He who calls you a man speaks truly ; he who calls 

you a fool calls you a man ; therefore he who 
calls you a fool speaks truly. 

64. Who is most hungry eats most ; who eats least is 

most hungry ; therefore who eats least eats most. 

65. What produces intoxication should be prohibited • 



320 QUESTIONS AND EXERCISES. . 

the use of spirituous liquors causes intoxication ; 
therefore the use of spirituous liquors should be 
prohibited. 

66. What we eat grew in the fields ; loaves of bread 

are what we eat ; therefore loaves of bread grew 
in the fields. 

67. If light consisted of material particles it would 

possess momentum ; it cannot therefore consist 
of material particles, for it does not possess 
momentum. 

68. Everything is allowed by law which is morally 

right ; indulgence in pleasures is allowed by law ; 
therefore indulgence in pleasures is morally right, 

69. All the trees in the park make a thick shade ; this 

is one of them, therefore this tree makes a thick 
shade. 

70. All visible bodies shine by their own or by re- 

flected light. The moon does not shine by its 
own, therefore it shines by reflected light ; but 
the sun shines by its own light, therefore it cannot 
shine by reflected light. 

71. Honesty deserves reward ; and a negro is a fellow- 

creature ; therefore, an honest negro is a fellow- 
creature deserving of reward. 

72. Nearly all the satellites revolve round their planets 

from v/est to east ; the moon is a satellite; there- 
fore it revolves round its planet from west to east. 

73. Italy is a Catholic country and abounds in beg- 

gars; France is also a Catholic country, and 
therefore abounds in beggars, 

74. Every law is either useless or it occasions hurt to 

some person ; now a law that is useless ought to 
be abolished ; and so ought every law that occa- 
sions hurt; therefore every law ought to be 
abolished. 



QUESTIONS AND EXERCISES. 321 

/5. The end of a thing is its perfection ; death is the 
end of life ; therefore death is the perfection of 
life. 

76. When we hear that all the righteous people are 

happy, it is hard to avoid exclaiming, What ! are 
all the unhappy persons we see to be thought 
unrighteous ? 

77. I am offered a sum of money to assist this person 

in gaining the office he desires ; to assist a 
person is to do him good, and no rule of morality 
forbids the doing of good; therefore no rule of 
morality forbids me to receive the sum of money 
for assisting the person. 

78. Ruminant animals are those which have cloven 

feet, and they usually have horns; the extinct 
animal which left this foot-print had a cloven 
foot; therefore it was a ruminant animal and 
had horns. Again, as no beasts of prey are rumi- 
nant animals it cannot have been a beast of prey. 

79. We must either gratify our vicious propensities, 

or resist them ; the former course will involve 
us in sin and misery; the latter requires self- 
denial; therefore we must either fall into sin 
and misery or practise self-denial. 

80. The stonemasons are benefitted by the masons' 

union ; the bricklayers by the bricklayers' union ; 
the hatmakers by the hatmakers' union; in 
short, every trade by its own union; therefore 
it is evident that if all workmen had unions all 
workmen would be benefitted thereby. 
61. Every moral aim requires the rational means of 
attaining it ; these means are the establishment 
of laws ; and as happiness is the moral aim of 
man it follows that the attainment of happiness 
requires the establishment of laws. 

21 



322 QUESTIONS AND EXERCISES. 

82. He that can swim needs not despair to fly ; for to 
swim is to fly in a grosser fluid, and to fly is to 
swim in a subtler. 

SS, The Helvetii, if they went through the country of 
the Sequani, were sure to meet with various 
difficulties ; and if they went through the Roman 
province, they were exposed to the danger of 
opposition from Caesar; but they were obliged 
io go one way or the other; therefore they were 
either sure of meeting with various difficulties, 
or exposed to the danger of opposition from 
Caesar. — De Bello Gallico, lib. I. 6. 

84. Riches are for spending, and spending for honour 

and good actions; therefore extraordinary ex- 
pense must be limited by the worth of the occa- 
sion. — Bacon, 

85. If light is not refracted near the surface of the 

moon, there cannot be any twilight ; but if the 
moon has no atmosphere light is not refracted 
near its surface; therefore if the moon has no 
atmosphere there cannot be any twihght. 

86. The preservation of society requires exchange; 

whatever requires exchange requires equitable 
valuation of property ; this requires the adoption 
of a common measure ; hence the preservation 
of society requires the adoption of a common 
meckwiure. 

87. The several species of brutes being created to 

prey upon one another proves that the human 
species were intended to prey upon them. 
68. The more correct the logic, the more certainly 
the conclusion will be wrong if the premises are 
false. Therefore where the premises are wholly 
uncertain, the best logician is the least safe 
guide. 



QUESTIONS AND EXERCISES, 323 

89. If our rulers could be trusted always to look to 

the best interests of their subjects, monarchy 
would be the best form of government ; but 
they cannot be trusted; therefore monarchy is 
not the best form of government ' - 

90. If men were prudent, they would act morally for 

their own good ; if benevolent, for the good of 
others. But many men will not act morally, 
either for their own good, or that of others ; such 
men, therefore, are not prudent or benevolent. 

91. He who bears arms at the command of the magis- 

trate does what is lawful for a Christian; the 
Swiss in the French service, and the British in 
the American service, bore arms at the command 
of the magistrate ; therefore they did what was 
lawful for a Christian. — Whately, 

92. A man that hath no virtue in himself ever envieth 

virtue in others ; for men's minds will either feed 
upon their own good or upon others^ evil ; and 
who wanteth the one will prey upon the other. — 
Bacon, \ 

93. The object of war is durable peace; therefore 

soldiers are the*-best peace-makers. 

94. Confidence in promises is essential to the inter- 

course of human life ; for without it the greatest 
part of our conduct would proceed upon chance. 
But there could be no confidence in promises, if 
men were not obliged to perform them ; the obli- 
gation, therefore, to perform promises is essential 
to the same ends and in tne same degree. 
S5. If the majority of those who use public-houses 
are prepared to close them, legislation is unne- 
cessary ; but if they are not prepared for such a 
measure, then to force it on them by outside 
pressur.e is both dangerous and unjust. 

21 — z 



324 QUESTIONS AND EXERCISES. 

96. He who believes himself to be always in the right 
in his opinion, lays claim to infallibility ; you 
always believe yourself to be in the right in youp 
opinion ; therefore you lay claim to infallibility. 
— Whately, 

07, If we never find skins except as the teguments of 
animals, we may safely conclude that animals 
cannot exist without skins. If colour cannot 
exist by itself, it follows that neither can any- 
thing that is coloured exist without colour. So, 
if language without thought is unreal, thought 
without language must also be so, 

©8. No soldiers should be brought into the field who 
are not well qualified to perform their part ; none 
but veterans are well qualified to perform their 
part ; therefore none but veterans should be 
brought into the field. — Whately, 

99. The ininimu7n visibile is the least magnitude which 
can be seen ; no part of it alone is visible, and 
yet all parts of it must affect the mind in order 
that it may be visible ; therefore, every part of 
it must affect the mind without being visible. 

100. The scarlet poppy belongs to the genus Papaver, 

of the natural order Papaveracese ; which again 
is part of the subclass Thalamiflorse, belonging 
to the great class of Dicotyledons. Hence the 
scarlet poppy is one of the Dicotyledons. 

101. Improbable events happen almost every day ; but 

what happens almost every day is a very pro- 
bable event ; therefore improbable events are 
very probable events. — Whately, 

Lesson XXI 1. — Quantification of the Predicate. 
2. What does the quantification of the predicate mean? 



QUESTIONS AND EXERCISES. 325 

2. Assign to each of the following propositions its 

proper symbol, and the symbol of its converse • 

(1) Knowledge is power. 

(2) Some rectangles are all squares. 

(3) Only the honest ultimately prosper. 

(4) Princes have but their titles for their glories. 

(5) In man there is nothing great but mind, 

(6) The end of philosophy is the detection of unity. 

3. Draw all the contrapositive propositions and imme- 

diate inferences you can from the following pro- 
positions : — 
(i) London is a great city. 

(2) London is the capital of England. 

(3) All ruminant animals are all cloven-footed ani- 

mals. 

(4) Some members of parliament are all the minis- 

ters. 

4. Write out in Hamilton's notation the moods Baroko 

Darapti, Felapton, Bokardo. 

Lesson XXIII. — Boole'' s System of Logic, 

1. Apply this system of inference to prove the syl- 

logisms on p. 141, in Cesare, and Camestres. 

2. Show that if all ^'s are not ^'s, then no ^'s are 

A's ; and that if all ^'s are all B^s, then all not 
A*s are all not ^'s. 

3. Develop e the term substance^ as regards the terms 

vegetable^ attimal^ organic ; then select the com- 
binations which agree with these premises : 
" What is vegetable is not animal but is or- 
ganic ; what is animal is organic." 

4. Test the validity of this argument : " Good always 

triumphs, and vice always fails ; therefore the 
victor cannot be wrong, nor the vanquished 
right." 



326 QUESTIONS AND EXERCISES. 

5. It is known of a certain class of things that — 
(i) Where the quality A is, B is not. 

(2) Where B is, and only where B is, C and D are. 
What can we infer from these premises of 
the class of things in which A is not pre- 
sent but C is present ? 

6. If all A's are ^'s ; all ^'s are C's; all C's are D's ; 

shew that all A^s are Z^'s, and that all notZ^'s are 
not A^s, 

Lesson 'XXIY. —Method. 

1. What is the supposed position of method accord- 

ing to former logical writers, and what are the 
rules of method ? 

2. Explain the expressions nobis notiora^ and notiora 

naturcB, 

3. Of what kind is the usual method of instruction ? 

4. Prove that analysis in extension is synthesis in in-? 

tension, using some of the series of terms in 
Question 6, Lesson v. as illustrations. 

5. Explain the exact meanings of the expressions a 

priori and a posteriori knowledge. 

6. To which kind belongs our knowledge of the fol- 

lowing facts 1 

(i) The light of the stars takes a long time to 
reach us. 

(2) Vaccination is a preservative against small-pox. 

(3) A meteor becomes heated in passing through the 

air. 

(4) There must be either some inhabitants or no 

inhabitants upon Jupiter. 

Lesson XXV. — Perfect Induction, 

I. Define and distinguish Deduction, Induction, and 
Traduction. 



QUESTIONS AND EXERCISES. 327 

2. Find an instance of reasoning in Traduction. 

3. Distinguish Perfect and Imperfect Induction. 

4. How does Mr Mill define Induction, and what is 

his opinion of Imperfect Induction? 

5. What is the use of Perfect Induction? 

6. Construct some instances of the inductive syllo- 

gism, and show that they may be thrown into a 
disjunctive form. 

Lesson XXVI. — Induction, Analogy and Example, 

1. From what circumstance arises the certainty and 

generality of reasoning in geometry ? 

2. Find other instances of certain and general reason- 

ing concerning the properties of numbers. 

3. Why are inductive conclusions concerning prime 

numbers uncertain and not general? 

4. Why is a single instance sometimes sufficient to 

warrant a universal conclusion, while in other cases 
the greatest possible number of concurring in- 
stances, without any exception, is not sufficient to 
warrant such a conclusion? 

5. What are the strict and ordinary meanings of the 

word analogy? 

6. Explain the use of Examples. 

7. Explain exactly the difference between analogical 

argument and ordinary induction. 

Lesson XXVII. — Observation afid Experiment. 

1. What is the false method of Science against 

which Bacon protested? 

2. Explain the exact meaning of Bacon's assertions, 

that man is the Servant and Interpreter of Nature, 
and that Knowledge is Power. 

3. How does experiment differ from observation ? 



|23 QUESTIONS AND EXERCISES. 

4. Classify the sciences according as they employ 

passive observation, experiment, or both. 

5. Name the chief points in which experiment is 

superior to mere observation. 

6. What is the principal precaution needful in obser- 

vation ? 

7. Explain how it is possible to anticipate nature and 

yet establish all conclusions upon the results of 
experience. 

Lessons XXVIII. and XXIX. — Methods of Induction, 

1. Define exactly what is meant by a cause of an 

event, and distinguish cause, occasion, antece^ 
dent, 

2. Point out all the causes concerned in the following 

phenomena : 
(i) The burning of a fire. 

(2) The ordinary growth of vegetables. 

(3) The cracking of a glass by hot water, 

3. State and explain in your own words Mr MilPis 

first three Canons of Inductive Method. 

4. Point out exactly how the Joint Method differs 

from the simple Method of Difference. 

5. Give some instances of simple experiments fulfil- 

ling completely the conditions of the Method of 
Difference. 

6. What can you infer from the following instances? 

Antecedents, Consequeftts. 

ABDE stqp 

BCD qsr 

BFG vqu 

ADE tsp 

HK xqw 

ABFG .pquv 

ABE ,pqt. 



QUESTIONS AND EXERCISES. 329 

7. (i) Friction alters the temperature of the bodies 

rubbed together. 

(2) The sun is supposed to move through space. 

(3) A ray of hght passing into or out of a denser 

medium is deflected. 
Point out the successive questions which would 
have to be decided in the investigation of the 
above phenomena. 

8. Find some simple instances of the homogeneous 

and heterogeneous intermixture of effects, and 
of the methods of concomitant variations and 
residues. 

9. Since 1842 there has been a great reform of the 

British tariff, and a great increase of British 
trade. Does this coincidence prove that the 
first circumstance is the cause of the second ? 
10. Supposing us to be unacquainted with the causes of 
the following phenomena, by what methods 
should we investigate each ? 

(i) The connection between the barometer and the 
weather. 

(2) A person poisoned at a meal 

(3) The connection between the hands of a clock. 

(4) The effect of the Gulf-stream upon the climate of 

Great Britain. 



Lesson XXX. — Empirical and Deductive Methods. 

1. Define Empirical Law, and find a few additional 

instances of such laws. 

2. What are the three steps of the Deductive Method ? 

3. Trace some of the successive steps in the progress 

of the theory of gravitation, showing that it was 
established by this method. 



330 QUESTIONS AND EXERCISES. 

Lesson XXXI. — Explanation^ &c, 

1. What do you mean by the explanation of a fact ? 

2. State the three ways in which a law of nature may 

be explained, and suggest some additional in- 
stances of each case. 

3. Define tendency. Do all causes consist only of 

tendencies, or can you find examples to the con- 
trary ? 

4. Give a definition of hypothesis. How may a valid 

be distinguished from an invalid hypothesis ? 

5. What place does hypothesis hold in the Deductive 

Method ? 

6. Explain the ambiguities of the words theory and 

fact. 

Lesson XXXI L — Classification. 

1. Define classification, and give the derivation of the 

word. 

2. What do you mean by important characters in 

classification ? 

3. State Dr WhewelPs criterion of a good natural 

arrangement. 

4. Distinguish between a natural and artificial system 

of classification. 

5. What do you mean by a characteristic quality ? Is 

it always an important quality ? 

6. Define abstraction, generalization, and colligation 

of facts. 

7. What are the characters of a notion properly abs- 

tracted? 

Lesson XXXIII. — Requisites of a Philosophical 

Language, 

le What are the three purposes for which we use 
language? 



QUESTIONS AND EXERCISES. 331 

2. What are the two chief requisites of a philosophical 

language ? 

3. By what considerations should we be guided in 

choosing between a new and old scientific term? 

4. Distinguish a Descriptive Terminology and a No- 

menclature ; separate the following terms ac- 
cording as they belong to one or the other: — 
Rose, Rosacese, Rose-like, Potassium, Alkaloid, 
Ruminant Animal, Ruminating, Ruby, Ruby-red. 
{. What does Mr Mill mean by the expression Na- 
tural Kind ? 



INDEX, 



AJTD CONCISS VOCABULARY OF LOGICAL AND PHILOSOPHI AL 

TERMS. 



Abacus; the logical, 199 

Abscissio Innniti (the cutting 
off of the infinite or negative part), 
the process by which we determine 
the position of an object in a system 
of classes, by successive comparison 
and rejection of those classes to which 
it does not belong. 

Absolute terms, i.e. non-relative 
terms, 25 ; sometimes used as name 
of non-connotative terms, 41 

Abstract terms, 20, 43 

Abstraction^ 285 

Accent; fallacy of, 174 

Accident; fallacy of, 176 ; the pre- 
dicable, 103 

Accidental definition is a defi- 
nition which assigns the properties 
of a species, or the accidents of an 
individual; it is more commonly 
called a Description. 

Acquired perceptions, 236 

Added determinants, inference 
by, 86 

Adequate knowledge, 56 

A dicto secundum quid^ &c., 
fallacy of, 176 

Adjectives, 21 

Adverbials, 93 

Affirmative propositions, 63 

Algebraic reasoning, 58, 219 

Ansbiguity of all, 20; oisoute^j^ 
of mitny old terms, 291 ; of terms in 
Political Economy, 292 

Ambiguous middle term, 130, 171 

Ampiiibology, fallacy of, 172 

Ampliative propositions, 69 

Analogue, a thing analogous to 
some other thing. 

Analysis, method of, 105 



Analogy, the cause of ambiguity 
35, 50; reasoning by, 226 — 8 

Analytics, (ra ' hvaXvTiKd,) the titl« 
given in the second century to por- 
tions of the Organon, or Logical 
Treatises of Aristotle; they were 
distinguished as the Prior and Pos- 
terior Analytics. 

Analytic syllogism, a syllogism 
in which the conclusion is placed 
first, the premises followjrg as the 
reasons. See Synthetic syllogism ^ 
the distinction is unimportant. 

Antecedent, of a hypothetical prfi 
position, 160; of an event, 240 

Anticipation of nature, 229 

Antinomy (aVrl, against; vofxo^, 
law), the opposition of one law or rule 
to another. Kant. 

A posteriori knowledge, 208 

A priori knowledge, 208 

Arbor Forpli3rriana, see Tree of 

Porphyry, 

Argument, (Latin, argus, from 
cipyb?, clear, manifest, ) the process of 
reasoning, the shewing or proving 
that which is doubtful by that which 
is known. See Infere7tce. The mid- 
dle term of a syllogism is sometimes 
called specially the argnjneJit. 

Argumentum a fortiori, an 
argument in which we prove I hat 
the case in question is more strong 
or probable than one already con- 
ceded to be sufficiently so. 

Argumentum ad hominem^ 

178 
Argumentum ad judicium^ 

an appeal to the common sense el 
mankind. 



INDEX. 



333 



Argumentum ad Ignoranti- 

am^ an argument founded on the 
ignorance of adversaries. 
Areuxnentum ad populum^ 

179 
Argamentum ad verecun- 
diani; an ax>peal to our respect for 
some great authority. 
Arguxnentuzn ex concesso^ 
a proof derived from a proposition 
already conceded. 
Aristotle ^s Bicta> 123 
Art and Science, distinction of, 7 
Artificial Classification, 284 
Assertion^ {ad^ to; sero^ to join,) 
a statement or proposition, affirma- 
tive or negative. 
Association of ideas, [associo, to 
accompany; sochts, a companion,) 
the natural connection existing in 
the mind between impressions which 
have previously coexisted, or which 
are similar. Any idea tends to bring 
into the mind its associated ideas, in 
accordance with the two great laws 
of association, the Law of Conti- 
guity, and the Law of Similarity. 
Assumption^ {assumo^ to take for 
granted,) any proposition taken as 
the basis of argument; in a special 
sense, the minor premise of a cate- 
gorical syllogism. 
Attribute^ {attribtiOy to jive or 
ascribe to,) a quality or circumstance 
which may be affirmed (or denied) 
of a thing; opposed to Substance^ 
which see. 
Attribute in grammar, 92 
Attributive term, i. e. Comioiative 

tenrty 41 
Axiom^ defininition of, 125 

Baconian method, 255; Philoso" 1 

phy, 229 
Barbara^ Celarent, &c,, 145 
Begging the Question, 179 
Belief* assent to a proposition, ad- 
milting of any degree of strength, 
from the slightest probability to the 
fullest certainty ; see Probability. 
Bentliam, George, new system of 

Logic, 187 
Boole, George, his system of Logic, 
191 ; his Laws of Thought, 197 ; 
his logical works, 201 



Canons of syllogism^ 121 — 2; Hamil- 
ton's supreme Canon, 189 

Canons of JMill's Inductive Methods, 
First, 240 ; Second, 242 ; Third, 245 ; 
Fourth, 252; Fifth, 249 

Categorematic words, 18 

Categorical propositions, 63 

Categories, the sujntna generUy or 
most extensive classes into which 
things can be distributed ; they are 
ten in number, as follows : 

OifcrCa, Substance ; UocroVy Quan- 
tity ; UoLOVy Quality ; IIpo? rt, Re- 
lation ; Iloielv, Action ; Ild<rx^eLV, 
Passion, or suffering ; IIov, Place ; 
IIoTe, I'ime ; Keta^at, Position ; 
"Exeit', Habit or condition. 

Everything which can be affirmed 
must come under one or other of these 
highest predicates, which were de- 
scribed in the first treatise of Aris- 
totle's Organojty called the Catego- 
ries. 

Cause, meaning of, 239 

Aristotle distinguished four kinds 
of causes for the existence of a thing 
— I. The Material Cause, the sub- 
1. stance or matter composing it ; 2. 
The Formal Cause, the pattern, type 
or design, according to which it is 
shaped ; 3. The Efficient Cause, the 
force employed in shaping it ; 4. 
The Final Cause, the end, motive 
or purpose of the work. 

Chance, ignorance of the causes 
which are in action ; see Probability. 

Character, derivation of the word, 
46 

Characteristics, 285 

CircUlUS in definiendo, iic, 1x4 

CirculUS in probando, 179 

Clearness of knowledge, 54 

Cognition, {cognosco, to know,) 
knowledge, or the action of nimd in 
acquiring knowledge. 

Colligation of Facts, Dr Whewell's 
expression for the mental union of 
facts by some suitable conception, 
see 286 

Collective terms, 19 

Combined or complete method of 
investigation, 258 

Comparison, [com, together ; paTy 
equal or like,) the action of mind by 
which we judge whether two object 



334 



INDEX. 



of thought are the same or different 
in certain points. See Jtcdgment. 
C033ipatlble terms are those which, 
though distinct, are not contradic- 
tory, and can therefore be affirmed 
of the same subject ; as '* large " and 
heavy ; " " bright-coloured ^ and 



** nauseous." 



Complex conception^ inference 

by, 87 
Complex sentence, 91 ; syllogism, 

158 
Composition of Causes^ the 

principle which is exemplified in all 
cases in which the joint effect of 
several causes is identical with the 
sum of their separate effects, y. S. 
Mill. See pp. 252, 265 

Composition^ fallacy of, 173 

Compound sentence, 90 

Compreliension of terms, sq&Ih- 
iensioK. 

Computation^ 127 

Concept^ that which is conceived, 
the result of the act of conception ; 
nearly synonymous with general no- 
tion, idea, thought. 

Conception [con^ together ; caJ>io^ 
to take). An ambiguous term, mean- 
ing properly the action of mind in 
which it takes several things toge- 
ther, so as to form a general notion ; 
or again, in which it forms "a men- 
tal image of the several attributes 
given in any word or combination of 
words." Mansel. 

Conceptualists^ 13 

Conclusion of .syllogism, 15, 127 ; 
weakened, 140 

Concrete terms, 20 

Conditional propositions, 62, 160 

Confusion of words, ambiguity 
from, 31 

Conjugate words, those which come 
from the same root or stock, as 
knowUy knoTuingy knowingly ^ know- 
ledge. 

Connotation of terms, 39, 41 ; 
ought to be exactly fixed, 290 

Consciousness^ the immediate 
knowledge which the mind has of 
its sensations and thoughts, and, in 
general, of all its present operations. 
Reid. 

Consectary = Corollary. 



Consequence^ the connection be- 
tween antecedent and consequent; 
but often used ambiguously for the 
latter. 

Consequent of a hypothetical pro- 
position, 161 

Consequent or effect of a cause, 
240 

Consequent; fallacy of the, i8r 

Conservation of energy, 263, 269 

Consilience of Inductions, the 
agreement of inductions derived 
from different and independent series 
of facts, as when we learn the mo- 
tion of the earth by entirely different 
modes of observation and reasoning* 
Whewell. 

Consistency of propositions, 78 

Consistent terms, see compatible 
terms. 

Contingent^ {contingo^ to touch,) 
that which may or may not happen ; 
opposed to the necessary and «V«- 
possible. 

Contingent matter, 80 

Continuity^ Law of, the principle 
that nothing can pass from one ex- 
treme to another without passing 
through all the intermediate degrees; 
motion, for instance, cannot be instan- 
taneously produced or destroyed. 

Contradiction^ Law of, 117, 193 

Contradictory terms, 24, zig; 
propositions, 76 

Contraposition^ conversion by, 
83, 186 

Converse fallacy of accident, 176 

Conversion of propositions, 82-- 85; 
with quantified predicate, 184 

Convertend, 82 

Coordinate propositions, 90 

Copula^ 16 

Corollary^ a proposition which fol- 
lows immediately from another which 
has been proved. 

Correction of observations, 253 

Correlative terms, 25 

Criterion (Kptry^ptoi', from KpCvu), to 
judge), any fact, rule, knowledge, 
or means requisite to the formation 
of a judgment which shall decide a 
doubtful question. 

Cross division^ 105 

Data^ (plural of datum^ that which 



INDEX. 



335 



is given,) the facts or assertions from 
which an inference is to be drawn. 

Deduction and Induction, 212 

Deductive or combined method, 
258, 272 

De facto^ what actually or really 
happens ; opposed to de jure, what 
ought to happen by law or right. 

Definition^ the logical process, 109, 
112; of logic, I 

Degree^ terms expressing, 24; ques- 
tions of, 120 

Demonstration^ {demonstro^ to 

point out,) strictly the pointing out 
the connection between premises and 
conclusion. The term is more ge- 
nerally used for any argument or 
reasoning regarded as proving an 
asserted conclusion. A demonstra- 
tion is either Direct or Indirect. In 
the latter case we prove the conclu- 
sion by disproving its contradictory, 
or shewing that the conclusion cannot 
be supposed untrue. 
Demonstrative Induction, 220 
"Hq nSorgan^s logical discoveries 

and writings, 190 
Denotation of terms, 39 
Deptb of a notion, see Intension, 
Derivatives from the root s.j>eCt 

sight, 52 
Descartes on Method, 116, 229 
Description^ sqq Accidental Dejl- 
nitiojt. 

Descriptive terminology, 292 
Destructive dilemma, 168; hypo- 
thetical syllogism, 162 — 4 
Desynonymization of terms, 49 
Determination, the distinguishing 
of parts of a genus by reunion of the 
genus and difference. See Division. 
Development of a term, 193 
Diagrams, of sentences, 93 — 7 ; of 
syllogisms, 129 — 133, 142; of pro- 
positions, 72 — 75 
Dialectic (StaAexTiK>) tckvtj, the art 
of discourse, from SioAeyeaSat, to 
discourse). The original name of 
Logic, perhaps invented by Plato ; 
also used to denote the Logic of 
probable Matter (Aristotle), the 
right use of Reason and Language, 
the Science of Being ; it is thus a 
highly ambiguous term. 
Diciiotomy, division by, 107, 193 



Dicta de omni et nullo, 123 
Difference, the predicabie, 99 
Differentiation of terms, 49 
Dilemma, 167 

Disbelief, the state of mind in which 
we are fully persuaded that some 
opinion is not true. J. S. Mill. It 
is equivalent to belief in the contra- 
dictory opinion or assertion, and is 
not to be confused with Doubt, which 
see. 

Discourse, or reasoning, 15 

Discovery, method of, 202 

Disjunctive, propositions, 62, 160; 
syllogism, 166, 194 

Distinct knowledge, 55 

Distribution of terms, 19, 74—5, 
82, 129 

Division, logical, 105 ; metaphysical 
108 ; fallacy of, 174 

Doubt, (dubito, to go two ways,) the 
state of mind in which we hesitate 
between two or more inconsistent 
opinions. See Disbelief. 

Drift of a proposition, the varying 
meaning which may be attributed to 
the same sentence according to ac- 
centuation. See Fallacy of accenty 
174— 5 

Smpiricism (eju.7retpta, experience), 
the doctrine of those who consider 
that all knowledge is derived merely 
from experience. 

Smpirical Law, 256 

Entbymeme, 153 

Epicheirema, 155 

Episyllogism, 155 

Equivocal terms, 29 

Equivocation, 30; causes of, 31; 
fallacy of, 171 

Essence, [essentia, from esse, to be,) 
" the very being of anything, where- 
by it is what it is." Locke. It is an 
ancient scholastic word, which can 
not be really defined, and should be 
banished from use. 

Essential propositions, 68 

Euler^s diagrams, 72 — 5, 129 — 133, 
142 

Evidence, {e, and videre^ to see,) 
literally the seeing of anything. 
The word now means any facts ap« 
prehended by the mind and mada 
the grounds of knowledge and beliefi 



33^ 



INDEX. 



examples^ use of, 237 

Exceptive propositions, 68 
Eiccladed xxuddle^ law of| zx7» 

119, 192 
Exclusive propositions, 68 
nxliaustive division^ 107, 192 
Zxperience^ 228 
Experimentum crucis, an ex- 
periment which decides between two 
rival theories, and shews which is to 
be adopted, as a finger-post shews 
which of two roads is to be taken. 
Explanation^ of facts, 264; of laws, 

265 
Explicative propositions, 68 
Exposita^ a proposition given to be 

treated by some logical process. 
Extension and intension, 37, 208 
Extensive Syllogism, 159 
Extremes of a proposition, are its 
ends or terms, the subject and predi- 
cate. 

Fact, 275 

Fallacy, purely logical, 170; semi- 
logical, 170 — 175 ; material, 176^ 
182 ; in hypothetical syllogism, 162 ; 
in dilemma, 168 
False cause, fallacy of, x8z 
False propositions, 70 
Figure of speech, fallacy of, 175 
Figures of the syllogism, 138 ; their 

uses, 143 
Form and matter of thought, a^ 
Fundamentum divisionis,io5 
Fundamentum relationis,the 
ground of relation, i.e. the series of 
events or circumstances which es- 
tablish a relation between two cor- 
relative terms. 
Fundamental principles of syllo- 
gism, 121 

Galenian, or 4tk figure of the syl- 
logism, 145 
General notions, 13 ; terms, 18 
Generalization, 286 ; of names, 

45 

Generic property, 102 

GenuS; 98 : generalissimum, 100 

Geometrical reasoning, 58, 218; 
Pascal on, 115 

Grammatical predicate, 88; sen- 
tence, 89 

Gravitation, theory of, 360 



Hamilton, Sir W. Method of No- 
tation, 187 
Serscliel, Sir J., on active aad 

passive observation, 234 

Xleterogeneous, loi ; intermix- 
ture of effects, 252 

Homogeneous, loi ; intermixture 
of effects, 252, 265 

Homologue^ whatever is homolo' 
gozis. 

Homology, a special term for the 
analogy existing between parts ol 
different plants and animals, as be- 
tween the wing of a bird and the 
fore leg of a quadruped, or between 
the scales of a fish and the feathers 
of a bird. 

Z3;omon3rmous terms, 30 

Hypothesis, 269, 270 

Hypothetical propositions, 62,160, 
syllogism, 161 — 2 

Idea (i5ea, cT5o9, image), a term used 
ambiguously, but generally equiva- 
lent to thought, notion, concept. 
Defined by Locke as ** Phantasm, 
notion, species, or whatever it is 
which the mind can be employed 
about in thinking." To have an idea 
of a thing is to think of that thing. 

Identity, law of, 117—8 

Idol (eidcuAof, ei6o;, image). Bacon's 
figurative name for the sources of 
error; he enumerated four kinds; 
Idols of the Tribe, which affect all 
people ; Idols of the Cave, which are 
peculiar to an individual ; of the 
Forum, which arise in the inter- 
course of men ; of the Theatre, which 
proceed from the systems of philoso- 
phers. 

Ignoratio Elenchi, 178 

Illation [illatmny past participle of 
infero^ to bring in). See Inference. 

Illative, that which can be inferred. 

Illicit process, of the minor term, 
131 ; of the major term, 132, 139 

Immediate inference, 85—7 

Imperfect figures of the syllo- 
gism, 145 

Imperfect Induction, 213 

Impossible matter, 80 

Inconsistent terms imply qualities 
which cannot coexist in the same 
thin£^. See compatible terms. 



INDEX. 



337 



Inconsistent propositions, 76 

Indefinite propositions, 65 

Indefinite or infinite term, is a ne- 
gative term which only marks an 
object by exclusion from a class. 

In design ate propositions. See In- 
definite propositions. 

Indirect demonstration. See De- 
monstration. 

Indirect inference^ method of, 
192 

Indirect reduction of the syllo- 
gism, 146, 148—9. 

Individual; what cannot be divided 
without losing its name and distinc- 
tive qualities, although generally 
capable of physical division or par- 
tition, which sec. 

Indiiction^ 212 

Inductive syllogism, 211, 214 

Inference^ defined, 81 ; immediate, 
85 — 87 ; mediate, 126 

Infima species^ 100 

Innate ideas, see a pHori trz€ths,izoZ 

Inseparable accident, 103 

Instances^ use of, 227 

Intension and extension of terms, 
37, 99, 208 ; law of relation, 40 

Intensive syllogism^ 159 

Intention, first and second, a dis- 
tinction between terms thus defined 
by Hobbes : — " Of the first ititett- 
tion are the names of things, a vtan^ 
sto7ie, &c. ; of the second are the 
names of names, and speeches, as 
universal, particular, genus, species^ 
syllogism, and the like." A term of 
the second intention expresses the 
mode in which the mind regards or 
classifies those of the first intention. 

Intermediate link^ explanation 
by, 267 

Intuitive knowledge, 57 

Inversion of subject and predicate, 

67 
Irrelevant conclusion, fallacy of, 

178 

Judg^ment^ 13 

Language^ the subject of logic, 10 
Language^ requisites of philoso- 
phical, 290 ; three purposes of, 287 
Laws of thought, 1, 117 : of nature, 
•239 



ZieibnitZ on Knowledge, 53 
Iienama (Aa/x^aKa>, to take or as< 
sume), a proposition, a premise 
granted ; in geometry, a preUminary 
proposition. 
Ziimitation^ conversion by, 82, 87 
laOgiC; derivation of name, 6 
IiOgical abacus, slate and machine, 

IiOgomacliy; 292 
Iiowest species^ 100 

IMEacliine, the logical, 199 
major; term, 128 ; premise, 129 
3^axiy questions; fallacy of, 182 
XMEaterial fallacies, 170, 176 
IMEatlieniatical induction, 220 
Scatter of thought, 4 ; of proposi- 
tions, 80 
IMtatter is defined by J. S. Mill as 
" the external cause to which we 
ascribe our sensations," or as P«^- 
manent Possibility of Sensation. 
IXEediate inference, 126 
Xaembra dividentia, the parts 
into which a class is divided ; tke 
constituent species of a genus. 
IMEetaphor; 50 
SSetaphysical division, 108 

BOietapIiysiCS [ja /xera Tci *u<rc*ccOi 
the works of Aristotle which fol- 
lowed or were studied after hia 
Physics. First Philosophy, or the 
so-called science of things in their 
own nature ; ontology or the science 
of Being. 

I^ethod {fxeOoSos, /u,6ra and 656?, 
way), mode, way or instrument of 
accomplishing an end. 

ISOietllod, the fourth part of logic, 
15, 201; Pascal on, 114; Descartes' 
Discourse on, 116; of indirect infer- 
ence, 192 

IKCetJlOds of Induction, Agreetnent, 
240 ; Difference, 242 ; of Experi- 
ment, 243 • Joint Method, 245 ; 
Residues, 252 ; Concomitant Varia- 
tions, 249 

]XIetonymy(^lera, and oi/o/i,a,name) 
grammatical name for the tranf^tfer 
of meaning of a word to a closely 
connected thing, as when we spcrik 
of the church, meaning the people iii 
it. See Transfer of meaning. 

IMCiddle Term; 126, 128 



22 



338 



INDEX. 



JSSill^ J. S., on Connotative terms, 
41 ; on Induction, 214 ; on Analogy 
and Induction, 227 ; on Observation, 
235 ; on Terminology and Nomen- 
clature, 294 
^inor term, 128 ; premise, 129 
I^Uemonic verses, Barbara^ &c., 

144 
]BSodal proposition, 69, 91 
3%2[OdUS; ponens^ 161 ; tollens, 162 
I^OdUS^ ponendo tollens, 166; tol- 

lendo ponerts^ 166 
2^00ds of the syllogism, 136; ac- 
cording to Hamilton, 188 

IPaiXie^ or term, 17 

IVatnral Classification^ 280 

BTatural Kinds^ 294 

Necessary matter, 80 

Ifecessity [ne, not ; and cesso^ to 
cease), that which always is and can- 
not but be. 

nregation^ conversion by, 83 

ZHTegative; terms, 22; propositions, 
63* 83; premises, fallacy of, 133 — 4 

ZVewton's experiments, 253, 259 

Zfomenclature^ 293 

N'ominal definitions, 112 

Nominalists; 13 

KTon causa pro causa^ 181 

Non sequitur, 181 

Notion {nosco, to know), the action 
of apprehending or taking note of 
the various qualities of an object ; or 
more commonly the result of that 
action. See Idea, Concept. 

Notiora naturae, 204 

Novum Organum, first aphor- 
isms of, 229 

Numerically definite syllogism, 
190 

Obj ect of verb, 93 

Objective, that which belongs to 
the object of thought, the non-ego \ 
opposed to Subjective, which see. 

Obscure knowledge, 54 

Observation, 231, 235 

Occasion of an event, the proximate 
cause, or last condition which is 
requisite to bring other causes into 
action ; 239 

Opposite terms, 24, 119 

Opposition of propositions, 78 

Oi^anon (opycu'oi/, Latin OrganutHy 



Instrument), a name for Aristotle's 
logical treatises, first generally used 
in the 15th century, implying that 
they may be regarded as an instru- 
ment to assist the mind. The name 
was adopted by Bacon for his Novum 
Organum. 

Paradox (irapdy 86^0^ contrary to 
opinion), an assertion contrary to 
common opinion, and which may or 
may not prove true ; often wrongly 
used to mean what is self-contradic- 
tory and absurd. 

FaralOg'ism (TrapaAoyc'^o/jtai, to rea- 
son wrongly), a purely logical fallacy, 
or breach of the rules of deductive 
logic. 

Parity of reasoning, an expression 
used to denote that when one case 
has been demonstrated, other simi- 
lar cases can be demonstrated by a 
like course of reasoning. 

Paronymons words, see Conjtt- 
gate words. 

Particular propositions, 63 — 6,72,79 

Particular premises, fallacy of, 135, 

151. 
Partition or physical division, 108 
Per accidens, conversion, 82 
Perfect Figure of the Syllogism, 

145 
Perfect knowledge^ characters 

of, 53 

Periodic changes, 250 

Peripatetic Philosophy (TreptTTareo), 
to walk about), the name usually 
given to the doctrines of Aristotie 
and his followers, who are said to 
have carried on their studies and 
discussions while walking about the 
halls and promenades of the Lyceum. 

Petitio Principii, 179 

Phenomenon, 240 

Philosophical language, re- 
quisites of, 290 

Physical definition assigns the 
parts into which a thing may bo 
separated by partition or physical 
division. 

Plurative propositions, 191 

Polylemma, an argument of the 
same form as a dilemma, but in which 
there are more than two alternatives. 

Porphyry, tree of; 103 



INDEX, 



339 



Port Royal IiOg^ic^ m, 157 

Positive terms, 22 

Post hoc, ergo propter boc^ 

181 

Postulate {postulatumj a thing de- 
manded), a proposition wliich is ne- 
cessarily demanded as a basis of ar- 
gunient ; in geometry, the postulates 
define the practical conditions re- 
quired. 

Predicables^ 98 

Predicaments {prcBdicamenta^ 
what can be predicated), see Cate- 
gories. 

Predicate^ 62, 88, 92 ; quantified, 
183 

Premise^ or Premiss, 15, 127 

Primary Laws of Thought, 117 

Principle {J>ri7icipiu7n, beginning), 
the first source of anything : some- 
times specially used to mean the 
major premise of a syllogism. 

Privative conception, inference 
by, 85 

Privative terms, 24 

Probability ;i quantity or degree of 
belief, or more truly, quantity of in- 
formation concerning an uncertain 
event, measured by the ratio of the 
number of cases favourable to the 
event to the total number of cases 
which are possible. 

Probability, of propositions, 70 ; of 
inductions, 223 

Problem [irpoPk-qiMa, that which is 
thrown down), an assertion put for- 
ward for proof or disproof. 

Proof, the assigning a reason or ar- 
gument for the support of a given 
proposition. 

Proper names, 18 

Property or propriutn^ 41, 102, 109 

Propositions, 10, 16 ; several kinds 
of, 60 ; affirmative and negative, 63 ; 
categorical, 63 ; conditional, 62, 160 ; 
disjunctive, 62, 160 ; essential or ex- 
plicative, 68; exclusive, exceptive, 
63* hypothetical, 62, 162; indefinite 
or indesignate, 65 ; modal, 69, 91 ; 
opposition of, 78; particular, 63-— 6, 
72, 79 ; pure, 69 ; plurative, 191 ; ir- 
regular, (i^ ; quality and quantity of, 

63 
Frosyllogism, 155 

Proximate genus, io8 



Quantification of predicate, 183 
Quantity of propositions, 63 ; ques« 

tions of quantity, 120 
Quaternio ternainorum, 170 

Ramean tree, see Tree of Por* 

phyry. 

Ratiocination, a name equivalent 
to Syllogism or Deduction, adopted 
by J. S. Mill. 

Realism, 13 

Reason {ration from reor^ to think), 
a term of wide and ambiguous mean- 
ing ; it has sometimes been specially 
used to denote the minor premise of 
a syllogism. 

Reasoning, or discourse, 15 

Record, language as instrument of, 
289 

Reductio ad absurdum or ad 
impossibile, an indirect demonstra- 
tion founded upon the impossibility 
of a contradictory supposition, 146 

Reduction of the syllogistic figures, 
145 ; of hypothetical to categorical 
syllogisms, 163 — 5 

Relation {relatum, past participle 
of re/ero, to bear back), any con- 
nection in thought or fact betweea 
two things, 21 

Relative terms, 25 

Residual phenomena, 254 

Residues, method of, 252 

Rules of the syllogism, 127 

Scliolastic Pliilosopliy, a ge- 
neral name for the systems of philo- 
sophy taught during the middle ages 
from the 9th to the i6th century, 
flourishing chiefly in the 13th and 
14th centuries. The subject was 
chiefly the logic of Aristotle, varied 
with theology, metaphysics, gram- 
mar, or rhetoric. 
Second Intention, see Intention, 
Secundi adjacentis, of the se- 
cond adjacent, an expression in in- 
correct Latin, applied to a gram- 
matical sentence or proposition con- 
taining only two parts, the subject 
and verb, without a distinct copiila. 
Self-contradictory terms, 193 
Semilogical fallacies, 171 
Sentence, grammatical, 61, 89 
Separable accident, 103 



340 



INDEX. 



Si^nificates of a term are thin^^s 
denoted or signified by it. 

Siiznilars; substitution of, 124, 200 

Simple^ apprehension, 11 ; conver- 
sion, 82, 184 

Singular; terms, 18 ; propositions,64 

Sopl^sm (o-6(^io-|uia, from aoibCa, wis- 
dom), a false argument ; the name 
often implies that a false argument 
is consciously used for deception. 

SoritdS; 156 

Specialization of names, 45, 48 

SpecieS; in logic, 98 ; in natural 
history, loi 

Subaltern^ propositions, 77; genera 
and species, 100 

Subalternans^ subaltern- 
ates, 77 

Subcontrary Propositions, 77 

Subject of a proposition, 62, 92 

Subjective^ that which belongs to 
the thinking subject, the e^o, or 
mind engaged in thought ; opposed 
to objective, which see. 

Subordinate propositions, 91 

Substance {suby under ; statis from 
stare y to stand), that which underlies 
and bears phenomena or attributes ; 
strictly speaking it is either mind or 
matter, but it is more commonly 
used in the material sense. 

Substitution of similars, 124, 200 

Subsumption [sub, under ; sutJto, 
to take or put), a name used by Sir 
W. Hamilton for the minor premise 
of a syllogism, because it brings or 
substanes a special case under the 
rule expressed in the major premise 
or sumption. 

Subsumption of a law is Mr 
Mill's expression for the third mode 
of explaining a law by shewing it to 
be a particular case of a more ge- 
neral law, o58 

Sufficient Reason^ Principle or 
Law of, 125 

Sui generis^ loi 

Summum genus^ 100 

Sumption {sumo, to take), Sir W. 
Hamilton's name for the major pre- 

. mise of a syllogism. 

Supposition^ 270 

Syllogism^ 10, 127; inductive, 211, 
214 

Symbolical knowledge, 57 



Syncategorematic words, z3 

Synthesis^ 205 

Syntlietic syllogism^ a syllo* 

gism in which the conclusion stands 
last ; see A nalytic syllogism. 
System^ {(jvimiixa, from avvCoTriixif 
to put together), a connected body 01 
knowledge. 

Tacit premise^ 153 
TautologOUS propositions, 69 
Tendency^ 266 
Terminology^ 292 

Terms; 10, 16, 17 

Tertii adjacentis^ of the third 

adjacent, an expression in incorrect 
Latin, applied to a grammatical sen- 
tence or proposition in which the 
subject, copula and predicate, aro 
all distinctly stated. 
Theory (^ewpta, contemplation), 
knowledge of principles, as opposed 
to practice; ambiguously used, see 

P- 274 
Thesis (deVi?, from riBrifxi, to place), 

an assertion or proposition which is 

put forth to be proved or supported 

by arguments. 
Thoughts on things, the object of 

logic, 10 
Totum divisumjF a class or notion 

which is divided into parts by a 

difference. 
Traduction^ 212 
Transfer of meaning of terms, 33 
Tree of Porphyry, 103 
Trilemma, an argument resem- 
bling a dilemma, but in which there 

arc three alternatives. 
Truisms, 69 
Truth, conformity of our knowledge 

"with the things known. 

Ultra-total distribution, xgi 
Uniformity of nature, 217 
Universal propositions, 63, 

66; affirmative, 71 ; negative, 73 
Univocal terms, 29 

Variations, method of, 249; pe- 
riodic, 250 
Verb, 88 

VTeakened conclusion, 140 
VTorse relation (Hamilton), 190 



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task thoroughly ^ and earned the gratitude of every student of botany '* 
—Spectator. 



SCIENTIFIC CLASS BOOKS. 



CHEMISTRY. 

LESSONS IN ELEMENTARY CHEMISTRY, INOR- 
GANIC AND ORGANIC. By Henry E. Roscoe, F.R.S., 
Professor of Chemistry in Owens College, Manchester. With 
numerous Illustrations. New Edition. i6mo, cloth, $i.io. 

^'As a standard general text-book it deserves to take a leading place, '^ 
— Spectator. 

* * We unhesitatingly pronounce it the best of all our elementary 
treatises on Chemistry.'" — Medical Times. 

THE OWENS COLLEGE JUNIOR COURSE OF PRACTICAL 
CHEMISTRY. By Francis Jones, Chemical Master in the 
Grammar School, Manchester. With Preface by Professor 
Roscoe. i8mo, cloth, 70 cents. 

A SERIES OF CHEMICAL PROBLEMS. For Use in Colleges 
and Schools. Adapted for the Preparation of Students for the 
Government, Science, ana Society of Arts Examinations. With 
a Preface by Professor Roscoe. i8mo, cloth, with Key, 50 cts. 

In the Preface Professor Roscoe says : * ^My expeAence has led me 
to feel more and more strongly that by no inethod can accuracy in a 
knowledge of chem^Hry be more surely sectored than by attention to the 
working of well-selected problems ^ and Dr. Thorpe's thorough ac- 
quaintance with the wants of the student is a sufficient guarantee that 
this selection has been carefully made. I intend largely to use these 
questions in my own classes ^ and I can confidently recommend them to 
all teachers and students of the science .^^ 

CHEMISTRY FOR STUDENTS. By A. W. Williamson, Phil. 
Doc, F. R. S., Professor of Chemistry, University College 
London. Second Edition, with Solutions. i6mo, cloth, $2.10. 

EXERCISES IN PRACTICAL CHEMISTRY. By A. G. Vernon 
Harcourt, M. a., F. R. S., and H. G. Mad an, M. A., Fellow 
of Queen's College, Oxford. Qualitative Exercises. i2mo, 
cloth, $1.90. 

''^An invaluable work for those who are beginning to learn practi- 
cally the beautiful science of Chemistry.'' — -Medical Press and Cir- 
cular. 

HEAT. 

AN ELEMENTARY TREATISE ON HEAT. By Bal- 
four Stewart, LL.D., F.R.S., Professor of Natural Philosophy 
at Owens College. Third Edition, enlarged. i6mo, 477 pp.,$?. 



SCIENTIFIC CLASS BOOKS, 



HSAT. — continued. 

To this edition have been added notices of such discoveries connected 
with ^^Heat'^ which have taken place since the second edition ivas 
published^ and also articles on " The Molecular Theory of Gases ^'' 
' * The Connection between the Two Elasticities and the Two Specific 
Heats'' ''^ The Velocity of Sound ^'' and ^''Graphical Representations of 
Physical Laws. " • 

LOGIC. 

ELEMENTARY LESSONS IN LOGIC ; Deductive and In- 
ductive, with copious Questions and Examples, and a Vocabu- 
lary of Logical Terms. By W. Stanley Jevons, M.A., Pro- 
fessor of Logic in Owens College, Manchester. New Edition, 
1877. i6mo, 90 cents. 

^^A Manual alike simple^ interesting^ and Scientific. "" — Athen^UM. 

* '// brings before the reader in a concise and very intelligible manner 
the whole body of recognized logical doctrines. Refers them to the great 
principles or so-called laws of thought from which they appear to be 
derived^ furnishes the student with a variety of examples^ and indicates 
the sources where he may find a full discussion of the subject 
treated.'" — Spectator. 

THE ELEMENTS OF DEDUCTIVE LOGIC, designed 
mainly for the use of Junior Students in the Universities. By 
T. Fowler, M.A., Fellow and Tutor of Lincoln College, Ox- 
ford. Sixth Edition, corrected and revised, with a Collection of 
Examples. i6mo, cloth, 90 cents. 

* ^Mr. Fowler appears to us to have accomplished his task skillfully 
and usefully. His book contains all the essential details of its subject, 
is clearly expressed, and embodies the result of much accurate 
thought. " — Guardian. 

THE ELEMENTS OF IN^ UCTIVE LOGIC, designed mainly 
for the use of Students in the Universities. By the same 
Author. Third Edition, corrected and revised. i6mo, 387 pp., 
cloth, $1.50. 

**^ most useful hand-book, mainly intended for University students , 
but which will be a convenient book, also, for those whose student days 
are over, but who wish to keep up with ?nore recent methods'' — LITER- 
ARY Churchman. 

MORAL PHILOSOPHY. 

HAND-BOOK OF MORAL PHILOSOPHY. By the Rev. 
Henry Calderwood, LL.D., Professor of Moral Philosophy, 
University of Edinburgh. Third Edition. i2mo, $1.50. 



SCIENTIFIC CLASS BOOKS. 



MORAL V^XLO^OV'B::^,'— continued, 

*'A compact and useful work, * * * will be an assistant to many 
students outside the author s own University , " — Guardian. 

'■^It isy we feel convinced, the best hand-book on the subject^ intellect- 
ually and morally , and does infinite credit to its author. " — Standard. 

NATURAL PHILOSOPHY. 

NATURAL PHILOSOPHY FOR BEGINNERS. With 
numerous Examples, By I. Todhunter, M.A., F.R.S. Part 
I. The Properties of Solid and Fluid Bodies. i8mo, cloth, 
90 cents. 

Part II. Sound, Light, and Heat. i8mo, cloth, 90 cents. 

From John M. Langley, Esq., University of Michigan : 

*^ I think this little book is thoroughly adapted for use in Grammar 
schools and others of a similar grade, and it will undoubtedly tend to 
hasten the day when elementary science shall be as universally taught 
as are now the * three K's,^ " 

From Charles R. Cross, Esq. Professor of Physics, Massachusetts 

Institute of Technology : 

"/^ would furnish a good basis of sound knowledge for some of us 
to build upon, instead of being obliged to cause our students to unlearn 
much which they suppose to be facts, ^^ 

From A. E. Dolbear, Esq., Professor of Physics, Tufts College, 

Mass, 

**/ have very carefully examined it, and find it to be a most excellent 
treatise, and one which might well supplant most of the text-books on 
that subject J^ 

** Perspicuous language, vigorous investigations, scrutiny of dif- 
ficulties, and methodical treatment characterize Mr, Todhunter^s 
works,'' — Civil Engineer. 

THE ELEMENTS OF NATURAL PHILOSOPHY. By Pro- 
fessor Sir W. Thomson, and P. G. Tait. Part I. 8vo, 
cloth, $2.50. 

LESSONS IN ELEMENTARY PHYSICS. By Balfour Stewart, 
F.R.S., Professor of Natural Philosophy in Owens College, 
Manchester. With numerous Illustrations. New Edition. 
i6mo, $1.10. 

** The active agents — heat, light, electricity, etc, — are regarded as 
varieties of energy, and the work is so arranged that their relation to 
one another, looked at in this light, and the paramount importance 
of the laws of energy, are clearly brought out. The volume contains 
all the necessary illustrations,'' — The Educational Times. 



SCIENTIFIC CLASS BOOKS, 



PHYSICAL GEOGRAPHY. 

ELEMENTARY LESSONS IN PHYSICAL GEOGRAPHY. 

By Archibald Geikie, F.R.S., Professor of Geology, etc., 
Edinburgh. With numerous Illustrations and Colored Maps. 
i6mo, $i.io. Questions on, 40 cents. 

^*' Anything more different from and more superior to the ordinary 
school-book it is im,possible to imagine. Were text-books adopted on 
their merits we should expect to see this one supplant all others on 
Physical Geography,'' — Christian Union. 

* ' We heartily commend this little volume to all teachers and students 
of Physical Geography'' — NATIONAL Journal of Education. 

* * The subject is treated in such a manner as to engage the interest of 
the young student^ and to excite him to observations and investigations 
for himself," — HARTFORD COURANT. 

PHYSIOLOGY. 

LESSONS IN ELEMENTARY PHYSIOLOGY. By T. H. 

Huxley, F.R.S. With numerous Illustrations. New Edition. 
i8mo, cloth, $1.10, Questions on, 40 cents. 

This book describes and explains, in a series of graduated lessons ^ 
the principles of Human Physiology, or the Structure and Functions 
of the Human Body, 

* 'Pure gold throughout, " — Guardian. 

* * Unquestionably the clearest and most complete elementary treatise 
on this subject that we possess in any language," — Westminster 
Review. 

A COURSE OF ELEMENTARY PRACTICAL PHYSIOLOGY, 
By M. Foster, M.D., F.R.S. Assisted by J. M. Langley, 
B.A. i2mo, cloth, $1.50. 

* ' This work will prove of great value to the teacher of physiology, as 
an aid to the preparation of an eminently practical course of lectures 
and demonstrations of elementary experimental physiology. Its chief 
utility, however, will be to the intelligent student, who, armed with a 
dissecting case, a microscope, and the book, will be enabled to pass his 
summer vacation in a manner at once interesting and profitable,"— - 
Medical Record. 

A TEXT-BOOK OF PHYSIOLOGY. By M. Foster, M.A., 
M.D., F.R.S. Third Edition, revised. i2mo, cloth, 75 cents ; 
sheep, $1.50. 
** After a careful perusal of the entire work we can cordially recom- 
mend it, both to the student and to the practitioner, as being one of the 
best text-books of Physiology extant, the facts recorded being as reliable 



SCIENTIFIC CIASS BOOKS. 



THYSIOIjOOY.— continued. ■ 

as the reasonings are sounds while the arrangement and style are alike 
excellent.'" — London Lancet. 

* */ recommend it to my students as the latest ^ and in some respects the 
best^ Physiology in the English Language."" — From a Letter from 
Professor Burt G. Wilder. 

POLITICAL ECONOMY. 

MANUAL OF POLITICAL ECONOMY. By Henry Faw- 
GETT, M.P., University of Cambridge. Fifth Edition, revised 
and enlarged. i2mo, cloth, 663 pp., $2.65. 

^^ It for??is one of the best iniroductio7ts to the principles of the 
science, and its practical applications !^ — Daily News. 

*' Tlie book is written throughout with admirable force ^ clearness and 
brevity^ every important part of the subject being duly considered T — 
Examiner. 

POLITICAL ECONOMY FOR BEGINNERS. By Millicent 
G. Fawcett. New Edition. i8mo, 75 cents. 

'* We ca7tnot conceive a book more fitted for popularizing this science 
than the clear, compact and comprehensive treatise, for which we are 
indebted to Mrs. Fawcett.'"— J^KYLY NEWS. 

*' The relations of capital and labor have never been more simply or 
more clearly expounded'' — CONTEMPORARY Review. 

A MANUAL OF POLITICAL ECONOMY. By J. E. Thor- 
OLD Rogers, M.A., formerly Professor of Political Economy, 
Oxford. Second Edition, with Index. i6mo, cloth, $1.10. 

'* Political economy is not a subject of which, in these days, sensible 
men can afford to be ignorant. Much of the ignorance which p7'evails 
respecting it will be cut at the root, if the able manual of Mr. Rogers 
is used extensively in our schools and colleges. '^ — GUARDIAN. 

STEAM. 

AN ELEMENTARY TREATISE OF STEAM. By John 
Perry, B.E.; Whitworth Scholar, etc., late Lecturer in Physics 
at Clifton College. With numerous Woodcuts, Numerical Ex- 
amples and Exercises. i8mo, $1.10. 

*^Mr. Perry has, in this compact little volume, brought together an 
immense amount of informatiort, new told, regarding steam and its 
application, not the least of its merits being that it is suited to the 
capacities alike of the tyro in engineer ing science or the better grade of 
artisan."" — Iron. 



8 SCIENTIFIC CLASS BOOKS. 



M ATHEMATICAL WORKS, 



BY 



I. TODHUNTER, M.A,F.R.S. 

Of St. John's College, Cambridge. 



{ ( 



Mr. Todhunter is chiefly known to students of Mathematics as 
the author of a series of admirable Mathe?natical text-books^ which 
possess the rare qualities of being clear in style ^ and absolutely free 
from mistakes y typographical or other.'' — Saturday Review. 

THE ELEMENTS OF EUCLID. For the use of Colleges and 
Schools. i8mo, 90 cents. 

ALGEBRA FOR BEGINNERS. With numerous examples. 
i6mo, 75 cents. Key, i2mo, $1.75. 

MENSURATION FOR. BEGINNERS. With numerous examples. 
i8mo, 75 cents. 

MECHANICS FOR BEGINNERS. With numerous examples. 
i8mo, $1.10. Key, $1.75. 

TRIGONOMETRY FOR BEGINNERS. With numerous exam- 
ples. iSmo, 75 cents. Key, i2mo, $2.25, 

ALGEBRA. For the use of Colleges and Schools. Seventh Edition. 
With new Chapters. i2mo, $1.80. Key, $2.60. 

PLANE TRIGONOMETRY. For the use of Colleges and Schools. 
i2mo, $1.30. Key, $2.60. 

A TREATISE ON SPHERICAL TRIGONOMETRY. i2mo, 

$1.10. 
AN ELEMENTARY TREATISE ON THE THEORY OF 

EQUATIONS. Third Edition. i2mo, $1.80. 

PLANE COORDINATE GEOMETRY, as applied to the Straight 
Line and the Conic Sections. i2mo, $1.80. 

A TREATISE ON THE DIFFERENTIAL CALCULUS. With 
numerous examples. i2mo, $2.60. 

A TREATISE ON THE INTEGRAL CALCULUS AND ITS 
APPLICATIONS. Fourth Edition. i2mo, $2.60. 



Macmillan & Co., 2 2 Bond St., New York. 



